text:
\subsection{Cartan type}\label{sec:by-diagram-cartan}

In what follows $q\in \Bbbk^{\times}$ has order $N > 1$. All generalized Dynkin diagrams are
numbered from the left to the right and from bottom to top.
If a numbered display contains several equalities (or diagrams), they will be referred to
with roman letters from the left to the right.

\subsubsection{Type $A_{\theta}$, $\theta \ge 1$}\label{subsec:type-A}

For $N > 2$, the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_{\theta}$, with defining
relations
\begin{align*}
y_{ij} &= 0, \quad i < j - 1; & y_{iij} &= \lambda_{iij}, \quad \vert j - i\vert = 1.
\end{align*}
This algebra is nonzero, see \cite[Proposition 5.2]{AAG}.

If $N = 2$, then $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_{\theta}$, with defining
relations
\begin{align*}
y_{ij} &= \lambda_{ij}, \quad i < j - 1; & [y_{(i\, i+2)}, y_{i+1}]_c &= \nu_i.
\end{align*}
This algebra is nonzero, see \cite[Proposition 4.2]{AAG}.

\subsubsection{Type $B_{\theta}$, $\theta \ge 2$, $N> 2$}\label{subsec:type-B}

\noindent
$\diamond${$N>4$}\label{subsubsec:type-B-N>4}
The algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_{\theta}$, with defining
relations
\begin{align*}
y_{ij} &= 0, \quad i < j - 1; & y_{iij}&=\lambda_{iij}, \quad i < \theta, \, \vert j - i\vert = 1;  &
y_{\theta\theta\theta\theta-1}&=\lambda_{\theta\theta\theta\theta-1}.
\end{align*}
If $N>5$, then $\lambda_{\theta\theta\theta\theta-1}=\lambda_{iij}=0$ for all $i,j$, so $\widetilde{\mathcal E}_{\bq}=\widetilde{\mathfrak{B}}_{{\mathfrak q}}$.
For $N=5$ all these scalars are 0 again, unless ${\mathfrak q}=\left(\begin{smallmatrix} q^2&q\\mathbf{q}^2&q\end{smallmatrix}\right)$, in which case $\lambda_{\theta-1\theta-1\theta}$, $\lambda_{\theta\theta\theta\theta-1}$ can be non-zero. This algebra projects onto the corresponding algebra of type $B_2$, which is nonzero, see \texttt{b2.log}.

\noindent
$\diamond${$N=4$}\label{subsubsec:type-B-N=4}
The algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_{\theta}$, with defining
relations
\begin{align*}
y_{ij} &= \lambda_{ij}, \quad i < j - 1; & [y_{(i\, i+2)}, y_{i+1}]_c&=\nu_i, \quad i < \theta;  &
y_{\theta\theta\theta\theta-1}&=0.
\end{align*}
Notice that $\nu_{\theta-1}=0$ since $\chi_{\theta-2}\chi_{\theta-1}^2 \chi_{\theta} (g_{\theta-2}g_{\theta-1}^2 g_{\theta})=-q$. Also, $\lambda_{i\theta}=0$ for all $i\leq \theta-2$, so $\widetilde{\mathcal E}_{\bq}$ projects over $\widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ of type $A_{\theta-1}$.

\noindent
$\diamond${$N=3$}\label{subsubsec:type-B-N=3}
The algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_{\theta}$, with defining
relations
\begin{align*}
y_{ij} &= 0, \quad i+1 < j < \theta; & y_{iij}&=\lambda_{iij}, \quad i < \theta, \, \vert j - i\vert = 1; \\
y_{i\theta} &= \lambda_{i\theta}, \quad i+1 < \theta;   &
[y_{\theta\theta\theta-1\theta-2}&,y_{\theta\theta-1}]_c=0.
\end{align*}
Notice that $\lambda_{\theta-1 \, \theta-1 \, \theta}=0$. If $\lambda_{\theta-1 \, \theta-1 \, \theta-2}=\lambda_{\theta-2 \, \theta-2 \, \theta-1}=0$, then $\widetilde{\mathcal E}_{\bq}$ projects onto $\widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ of type $A_{\theta-2} \times A_1$. If either $\lambda_{\theta-1 \, \theta-1 \, \theta-2}\neq0$ or $\lambda_{\theta-2 \, \theta-2 \, \theta-1}\neq 0$, then
$\lambda_{\theta-2 \, \theta-2 \, \theta-3}=\lambda_{\theta-3 \, \theta-3 \, \theta-2}=0$; $\widetilde{\mathcal E}_{\bq}/ \langle y_{\theta-2 \, \theta-3} \rangle$ projects onto $\widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ of type $A_{\theta-3} \times B_3$, which is nonzero, see \texttt{b3.log}.

\subsubsection{Type $C_{\theta}$, $\theta \ge 3$, $N> 2$}\label{subsec:type-C}

\noindent
$\diamond${$N>3$}\label{subsubsec:type-C-N>3}
The algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_{\theta}$, with defining
relations
\begin{align*}
y_{ij} &= 0, \quad i < j - 1; &
y_{iij}&=0, \quad i \leq \theta-2, \, \vert j - i\vert = 1;
\\
y_{\theta-1\theta-1\theta-1\theta}&=\lambda_{\theta-1\theta-1\theta-1\theta}; &
y_{\theta\theta\theta-1}&=\lambda_{\theta\theta\theta-1}.
\end{align*}
If $N\neq 5$, then $\widetilde{\mathcal E}_{\bq}=\widetilde{\mathfrak{B}}_{{\mathfrak q}}$.
For $N=5$, $\boldsymbol\lambda=0$ again, unless ${\mathfrak q}=\left(\begin{smallmatrix} q & q^2 \\ q & q^2 \end{smallmatrix}\right)$, in which case $\lambda_{\theta-1\theta-1\theta-1\theta}$, $\lambda_{\theta\theta\theta-1}$ can be non-zero. Thus $\widetilde{\mathcal E}_{\bq}$ projects onto the corresponding algebra of type $B_2$, which is nonzero, see \texttt{b2.log}.

\noindent
$\diamond${$N=3$}\label{subsubsec:type-C-N=3}

The algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_{\theta}$, with defining
relations
\begin{align*}
y_{ij} &= 0, \quad i+1 < j <\theta; &
y_{iij}&=\lambda_{iij}, \quad j=i\pm 1, i<\theta-1, & y_{\theta\theta\theta-1}&=0;
\\
y_{i\theta} &= \lambda_{i\theta}, \quad i < \theta - 1; &
&[[ y_{(\theta-2\theta)}, y_{\theta-1}]_c, y_{\theta-1}]_c
=\lambda_{(\theta-2\theta)}.
\end{align*}
If $\lambda_{(\theta-2\theta)}=0$, then $\widetilde{\mathcal E}_{\bq}/\langle y_{\theta-1 \theta}\rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ of type $A_{\theta-1}\times A_1$. If $\lambda_{(\theta-2\theta)}\neq 0$, then we may restrict to rank 3 following \S \ref{subsubsec:casebycase} and ${\mathfrak q}=\left(\begin{smallmatrix}
\zeta & b\zeta^2 & b^{-3}\zeta^2 \\
 b^{-1} & \zeta & b \\
b^3\zeta & b^{-1}\zeta & \zeta^2
\end{smallmatrix}\right)$; this algebra is not zero by \texttt{c3.log}.

\subsubsection{Type $D_{\theta}$, $\theta \ge 4$}\label{subsec:type-D}

\noindent
$\diamond${$N > 2$}\label{subsubsec:type-D-N>2}
The algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_{\theta}$, with defining
relations
\begin{align*}
y_{\theta-1\theta} &=0; & y_{ij} &= 0, \quad i < j - 1, (i,j)\neq (\theta-2,\theta);  &
y_{\theta\theta\theta-2}&=\lambda_{\theta\theta\theta-2}; \\
&& y_{iij} &= \lambda_{iij}, \quad \vert j - i\vert = 1, i,j\neq \theta; &
y_{\theta-2\theta-2\theta}&=\lambda_{\theta-2\theta-2\theta}.
\end{align*}
If $N\neq 3$, then $\boldsymbol\lambda=0$, so $\widetilde{\mathcal E}_{\bq}=\widetilde{\mathfrak{B}}_{{\mathfrak q}}$.
If $N=3$, then either $\lambda_{\theta-2\theta-2\theta-1}= \lambda_{\theta-1\theta-1\theta-2}=0$ or else $\lambda_{\theta-2\theta-2\theta}= \lambda_{\theta\theta\theta-2}=0$ by \cite[Lemma 5.1 (2)]{AAG}; we may assume the last case, and $\widetilde{\mathcal E}_{\bq}/ \langle y_{\theta} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ of type $A_{\theta-1}$.

\noindent
$\diamond${$N = 2$}\label{subsubsec:type-D-N=2}
The algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_{\theta}$, with defining
relations
\begin{align*}
y_{ij} &= \lambda_{ij}, \quad i < j - 1, (i,j)\neq (\theta-2,\theta);  &  y_{\theta-1\theta} &=\lambda_{\theta-1\theta}; \\
  [y_{(ii+2)},& y_{i+1}]_c = \nu_i, \quad i\leq \theta-3; &
[y_{\theta-3\theta-2\theta},y_{\theta-2}]_c&=\nu_{\theta-2}'.
\end{align*}
By Diamond Lemma \cite{B - diamond} $\widetilde{\mathcal E}_{\bq}\neq 0$: the defining relations are as for type $A$, so the proof of \cite[Proposition 4.2]{AAG} applies to solve the ambiguities.

\subsubsection{Type $E_{\theta}$, $6 \le \theta \le 8$}\label{subsec:type-E}

The proof is analogous to the case $D_{\theta}$.

\subsubsection{Type $F_{4}$, $N> 2$}\label{subsec:type-F}

\noindent
$\diamond${$N>4$}\label{subsubsec:type-F-N>4}
The algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining
relations
\begin{align*}
y_{ij} &= 0, \quad i < j - 1; & y_{112}&=0, & y_{221}&=0, & y_{334}&=0, \\ y_{443}&=0, & y_{223}&=\lambda_{223}, &  y_{3332}&=\lambda_{3332}.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_1,y_4 \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ of type $B_2$.

\noindent
$\diamond${$N=4$}\label{subsubsec:type-F-N=4}
The algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining
relations
\begin{align*}
y_{ij} &= 0, \quad i < j - 1; & [y_{(13)}, y_{2}]_c&=0; & y_{3332}&=0  \\
y_{334}&=0; & y_{443}&=0.
\end{align*}
Hence, $\widetilde{\mathcal E}_{\bq}=\widetilde{\mathfrak{B}}_{{\mathfrak q}} \neq 0$.

\noindent
$\diamond${$N=3$}\label{subsubsec:type-F-N=3}
The algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining
relations
\begin{align*}
y_{ij} &= \lambda_{ij}, \quad i < j - 1; & [[y_{(24)}, y_3]_c, y_3]_c&=\lambda_{(24)};\\
y_{iij}&=\lambda_{iij}, \quad  j=i\pm 1, (i,j)\neq (3,2);   &
[y_{3321},y_{32}]_c&=\lambda_{321}.
\end{align*}
Notice that $\lambda_{223}=0$.
If $\lambda_{321}=\lambda_{(24)}=0$, then $\widetilde{\mathcal E}_{\bq}/ \langle y_{23} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ of type $A_2\times A_2$. If $\lambda_{321}\neq0$ but all the other $\lambda$'s are 0, then $\widetilde{\mathcal E}_{\bq}/ \langle y_{34} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ of type $B_3\times A_1$. If $\lambda_{(24)}\neq0$ but all the other $\lambda$'s are 0, then $\widetilde{\mathcal E}_{\bq}/ \langle y_{12} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ of type $C_3\times A_1$.  The general case follows by applying Lemma
\ref{lem:nonzero-k<l} twice.

\subsubsection{Type  $G_{2}$, $N > 3$}\label{subsec:type-G}
The algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_1, y_2$ with defining
relations
\begin{align*}
y_{112} &= \lambda_{1}; &y_{22221}&=\lambda_{2}
\end{align*}
with $\lambda_1=\lambda_2=0$ unless $N=7$ and ${\mathfrak q}=\left(\begin{smallmatrix}
q&q^3\\mathbf{q}&q^3
\end{smallmatrix}\right)$. This algebra is nonzero, see \texttt{g2.log}.

\subsection{Standard type}\label{sec:by-diagram-standard}

\subsubsection{Standard type $B_{\theta}$, $\theta \ge 2$, $\zeta \in
\mathrm{G}'_3$}\label{subsec:type-B-standard}
The algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_{\theta}$, with defining
relations
\begin{align*}
y_{ij} &= \lambda_{ij}, \quad i < j - 1; &  [y_{(i-1i+1)},y_i]_c&=\nu_i, \quad
q_{ii}=-1;  \\
 y_{iii\pm1} &= 0, \quad q_{ii}=-\zeta^{\pm
1}; & [y_{\theta\theta\theta-1\theta-2},y_{\theta\theta-1}]_c&=0;  \\
 y_\theta^3&=\mu_\theta;  & [y_{\theta\theta\theta-1},y_{\theta\theta-1}]_c
 &=\lambda_{\theta\theta-1},
\quad q_{\theta-1\theta-1}=-1; \\
  y_i^2&=\mu_i, \quad
q_{ii}=-1.
\end{align*}
As $\chi_{\theta-1}\chi_\theta^2\chi_{\theta+1} (g_{\theta-1}g_\theta^2 g_{\theta+1})\neq 1$, $\nu_{\theta-1}=0$. Hence, if either $q_{\theta-1\theta-1}\neq -1$ or $q_{\theta-1\theta-1}=-1$ but $\lambda_{\theta\theta-1}=0$, then $\widetilde{\mathcal E}_{\bq}/ \langle y_{\theta-1\theta} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, one of super type $A$ and another of type $A_1$.

If $q_{\theta-1\theta-1}=-1$ and $\lambda_{\theta\theta-1}\neq 0$, then either $q_{\theta-2\theta-2}\neq 1$ or else $q_{\theta-2\theta-2}\neq 1$, $\nu_{\theta-2}=-1$. In both cases, $\widetilde{\mathcal E}_{\bq}/ \langle y_{\theta-2\theta-1} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, one of super type $A$ and another of type $B_2$ with matrix
$\left(\begin{smallmatrix} -1 & \zeta \\ -1 & \zeta
\end{smallmatrix}\right)$; this algebra is nonzero, see \texttt{b2st.log}.

\subsubsection{Standard type $G_{2}$, $\zeta \in \mathrm{G}'_8$}\label{subsec:type-G-st}
The Weyl groupoid has three objects.

For $\xymatrix{ \overset{\,\, \zeta^2}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  &
\overset{\overline{\zeta}}{\underset{\ }{\circ}}}$, $\widetilde{\mathcal E}_{\bq}$ is generated by $y_1, y_2$ with defining relations
\begin{align*}
y_1^4&=\lambda_1; & y_{221}&=\lambda_2; & [y_{1112},y_{112}]&=\lambda_3+2\lambda_2(1+q)^2y_2y_{12},
\end{align*}
with $\lambda_1=\lambda_2=\lambda_3=0$ unless ${\mathfrak q}=\left(\begin{smallmatrix}
\zeta^2&\zeta^{-1}\\\zeta^2&\zeta^{-1}
\end{smallmatrix}\right)$; see \texttt{g2-st-a1.log} for the deformation of the generalized quantum Serre relation.
This algebra is nonzero, see \texttt{g2-st-a.log}.

For $\xymatrix{ \overset{\,\, \zeta^2}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta^3}  &
\overset{-1}{\underset{\ }{\circ}}}$, $\widetilde{\mathcal E}_{\bq}$ is generated by $y_1, y_2$ with defining relations
\begin{align*}
y_1^4&=\lambda_1; & y_2^2&=\lambda_2; & [y_1,y_{3\alpha_1+2\alpha_2}]_c+\frac{q_{12}}{1-\zeta}y_{112}^2&=0,
\end{align*}
with $\lambda_1\lambda_2=0$. This algebra is nonzero, using \cite[Lemma 5.16]{AAGMV}. The last relation is not primitive: it remains undeformed by direct computation.

For $\xymatrix{ \overset{\zeta}{\underset{\ }{\circ}} \ar
 @{-}[r]^{\zeta^5}  & \overset{-1}{\underset{\ }{\circ}}}$, $\widetilde{\mathcal E}_{\bq}$ is generated by $y_1, y_2$ with defining relations
\begin{align*}
y_{11112}&=\lambda_1; & y_2^2&=\lambda_2, & [y_{112},y_{12}]_c,y_{12}]_c&=\lambda_3;
\end{align*}
with $\lambda_1=\lambda_2=\lambda_3=0$ unless ${\mathfrak q}=\left(\begin{smallmatrix}
\zeta&-1\\\zeta&-1
\end{smallmatrix}\right)$.
This algebra is nonzero, see \texttt{g2-st-c.log}.

\subsection{Super type}\label{sec:by-diagram-super}

\subsubsection{Type $\supera{j-1}{\theta - j}$, $\theta \ge 1$, $1\le j\le \theta$, $N >
2$}\label{subsec:type-A-super}

The algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_{\theta}$, with defining
relations
\begin{align*}
y_{ij} &= \lambda_{ij}, \quad i < j - 1; & y_{iii\pm1} &= \lambda_{iii\pm1}, \quad
q_{ii}\neq-1;  \\
 y_i^2&=0, \quad q_{ii}=-1;  & [y_{(i-1i+1)},y_i]_c&=\nu_i, \quad
q_{ii}=-1.
\end{align*}
By induction on $\theta$. If $\theta=2$, then there are 3 diagrams.
\begin{description}[leftmargin=*]
\item[$\xymatrix{ \overset{q}{\underset{\ }{\circ}} \ar@{-}[r]^{q^{-1}}  & \overset{q}{\underset{\ }{\circ}}}$] It is of Cartan type, already solved.
\item[$\xymatrix{\overset{-1}{\underset{\ }{\circ}} \ar@{-}[r]^{q}  & \overset{-1}{\underset{\ }{\circ}}}$] $y_i^2=\mu_i$, $i=1,2$. Hence \cite[Lemma 5.16]{AAGMV} applies.
\item[$\xymatrix{ \overset{q}{\underset{\ }{\circ}} \ar@{-}[r]^{q^{-1}}  & \overset{-1}{\underset{\ }{\circ}}}$] $y_{112}=\lambda_{112}$, $y_2^2=\mu_2$. If $\lambda_{112}=0$, then \cite[Lemma 5.16]{AAGMV} applies. Otherwise, ${\mathfrak q}=\left(\begin{smallmatrix} q & -1 \\ q & -1 \end{smallmatrix}\right)$, $q\in\mathrm{G}_4'$; $\widetilde{\mathcal E}_{\bq}\neq 0$ by \texttt{a2-super.log}.
\end{description}

For $\theta=3$, we have six cases:
\begin{description}[leftmargin=*]
	\item[$\xymatrix{ \overset{q}{\underset{\ }{\circ}} \ar@{-}[r]^{q^{-1}}  & \overset{q}{\underset{\ }{\circ}} \ar@{-}[r]^{q^{-1}}  & \overset{q}{\underset{\ }{\circ}}}$] It is of Cartan type, already solved.
	\item[$\xymatrix{ \overset{q}{\underset{\ }{\circ}} \ar@{-}[r]^{q^{-1}}  & \overset{q}{\underset{\ }{\circ}} \ar@{-}[r]^{q^{-1}}  & \overset{-1}{\underset{\ }{\circ}} }$] If $\lambda_{223}=0$, then we quotient by $y_{23}$. If $\lambda_{223}\neq 0$, then $q\in\mathrm{G}_4'$, so $\lambda_{112}=\lambda_{221}=0$ and we quotient by $y_{12}$.
	\item[$\xymatrix{ \overset{q}{\underset{\ }{\circ}} \ar@{-}[r]^{q^{-1}}  & \overset{-1}{\underset{\ }{\circ}} \ar@{-}[r]^{q}  & \overset{q^{-1}}{\underset{\ }{\circ}}}$] First assume $\nu_2=0$. If $\lambda_{112}\lambda_{332}=0$, then it projects over a rank 2 case. Otherwise, ${\mathfrak q}=\left(\begin{smallmatrix} q & -1 & b \\ q & -1 & -q \\ -b & -1 & -q
	\end{smallmatrix}\right)$, $b=\pm q\in\mathrm{G}_4'$; this algebra is nonzero,
	see \texttt{a3-super-1.log}. If $\nu_2\neq 0$, then ${\mathfrak q}=\left(\begin{smallmatrix} q & a & q^{-1}a^{-2} \\ q^{-1}a^{-1} & -1 & qa \\ qa^2 & a^{-1} & q^{-1} \end{smallmatrix}\right)$, $a\neq 0$; this algebra is nonzero, see \texttt{a3-super-2.log}. The general case follows by Lemma \ref{lem:nonzero-k<l}.
	\item[$\xymatrix{\overset{-1}{\underset{\ }{\circ}} \ar@{-}[r]^{q}  & \overset{-1}{\underset{\ }{\circ}} \ar@{-}[r]^{q^{-1}}  & \overset{q}{\underset{\ }{\circ}}}$] as $\nu_2=0$, we quotient by $y_{12}$.
	\item[$\xymatrix{\overset{-1}{\underset{\ }{\circ}} \ar@{-}[r]^{q^{-1}}  & \overset{q}{\underset{\ }{\circ}} \ar@{-}[r]^{q^{-1}}  & \overset{-1}{\underset{\ }{\circ}}}$] If $\lambda_{221}\lambda_{223}=0$, then we quotient either by $y_{12}$ or $y_{23}$. Otherwise, ${\mathfrak q}=\left(\begin{smallmatrix} -1 & q & -1 \\ -1 & q & -1 \\ -1 & q & -1
	\end{smallmatrix}\right)$, $q\in\mathrm{G}_4'$; this algebra is nonzero, see \texttt{a3-super-3.log}.
	\item[$\xymatrix{\overset{-1}{\underset{\ }{\circ}} \ar@{-}[r]^{q^{-1}}  & \overset{-1}{\underset{\ }{\circ}} \ar@{-}[r]^{q}  & \overset{-1}{\underset{\ }{\circ}}}$] If $\nu_2=0$, then we quotient by $y_{12}$. If $\nu_2\neq 0$, then ${\mathfrak q}=\left(\begin{smallmatrix} -1 & a & -a^{-2} \\ q^{-1}a^{-1} & -1 & qa \\ -a^2 & a^{-1} & -1 \end{smallmatrix}\right)$, $a\neq 0$; this algebra is nonzero, see \texttt{a3-super-4.log}.
\end{description}

For the inductive step, if the $\lambda$'s involving $y_{\theta}$, $y_{\theta+1}$ are 0, then $\widetilde{\mathcal E}_{\bq}/ \langle y_{\theta\theta+1} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, of super type $A$ and $A_1$. If the $\lambda$'s involving $y_{\theta}$ and $y_{\theta+1}$ are the unique non-zero scalars, then it projects over the rank 2 or rank 3 cases. The general case follows by Lemma \ref{lem:nonzero-k<l}.

\subsubsection{Type $\superb{m}{n}$, $q\notin\mathrm{G}_4$}\label{subsec:type-B-super}

\noindent
$\diamond${$N>4$}\label{subsubsec:type-B-super-N>4}
The algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_{\theta}$, with defining
relations
\begin{align*}
y_{ij} &= \lambda_{ij}, \, i < j - 1; \qquad y_i^2=\mu_i, \, q_{ii}=-1;   &  y_{\theta\theta\theta\theta-1}&=\lambda_{\theta\theta\theta\theta-1}; \\
y_{iij}&=\lambda_{iij}, \, i < \theta, |j-i|=1, q_{ii}\neq -1; & [y_{(i-1\, i+1)},y_i]_c&=\nu_i, \, q_{ii}= -1.
\end{align*}
First assume $q_{\theta-1\theta-1}\neq -1$. If $\lambda_{\theta\theta\theta\theta-1} =\lambda_{\theta-1\theta-1\theta}=0$, then we consider the quotient $\widetilde{\mathcal E}_{\bq}/ \langle y_{\theta-1\theta} \rangle$ following \S \ref{subsubsec:cut}. If either $\lambda_{\theta\theta\theta\theta-1}\neq 0$ or $\lambda_{\theta-1\theta-1\theta}\neq 0$, then $q\in\mathrm{G}_5'$ and $\nu_{\theta-2}=0$; hence
$\widetilde{\mathcal E}_{\bq}/ \langle y_{\theta-2\theta-1} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, one of super type $A$ and another of Cartan type $B_2$.

When $q_{\theta-1\theta-1}=-1$, $\nu_{\theta-1}=0$ since $\chi_{\theta-2}\chi_{\theta-1}^2\chi_\theta\neq \epsilon$. If  $\lambda_{\theta\theta\theta\theta-1}=0$, then again we consider $\widetilde{\mathcal E}_{\bq}/ \langle y_{\theta-1\theta} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$.
If $\lambda_{\theta\theta\theta\theta-1}\neq 0$, then $q\in\mathrm{G}_6'$ and $\nu_{\theta-2}=0$; hence
$\widetilde{\mathcal E}_{\bq}/ \langle y_{\theta-2\theta-1} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, one of super type $A$ and another of Cartan type $B_2$ with matrix $\left(\begin{smallmatrix} -1 & q \\ -1 & q \end{smallmatrix}\right)$; this algebra is nonzero, see \texttt{b2supera.log}.

\noindent
$\diamond${$N=3$}\label{subsubsec:type-B-super-N=3}
The algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_{\theta}$, with defining
relations
\begin{align*}
y_{ij} &= \lambda_{ij}, \quad i < j - 1; &
[y_{(i-1i+1)},y_i]_c&=\nu_i, \quad q_{ii}=-1;  \\
y_{iii\pm1} &= \lambda_{iii\pm1}, \quad q_{ii}\neq -1; &
[y_{\theta\theta\theta-1},y_{\theta\theta-1}]_c&=\lambda_{\theta\theta-1}, \quad q_{\theta-1\theta-1}=-1;\\
[y_{\theta\theta\theta-1\theta-2},&y_{\theta\theta-1}]_c
=\lambda_{\theta\theta-1\theta-2};  &
y_i^2&=\mu_i, \quad q_{ii}=-1.
\end{align*}
First assume $\lambda_{\theta\theta-1\theta-2}\neq 0$. Then $q_{ii}=q^2$ for $i=\theta-2,\theta-1$. Moreover, $\lambda_{\theta-2\theta-2\theta-3}=0$, $\nu_{\theta-3}=0$ if $q_{\theta-3\theta-3}=-1$, $\lambda_{\theta-3\theta-3\theta-2}=0$ if $q_{\theta-3\theta-3}=q^2$;  hence
$\widetilde{\mathcal E}_{\bq}/ \langle y_{\theta-3\theta-2} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, one of super type $A$ and another of Cartan type $B_3$.

Now assume $\lambda_{\theta\theta-1\theta-2}=0$. If either $q_{\theta-1\theta-1}=q^2$ or $q_{\theta-1\theta-1}=-1$, $\lambda_{\theta\theta-1}=0$, then $\widetilde{\mathcal E}_{\bq}/ \langle y_{\theta-1\theta} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, one of super type $A$ and another of type $A_1$. If $q_{\theta-1\theta-1}=-1$, $\lambda_{\theta\theta-1}\neq 0$, then $\nu_{\theta-1}=0$, $\nu_{\theta-2}=0$ if $q_{\theta-2\theta-2}=-1$, $\lambda_{\theta-2\theta-2\theta-1}=0$ if $q_{\theta-2\theta-2}\neq -1$;
hence $\widetilde{\mathcal E}_{\bq} / \langle y_{\theta-2\theta-1} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, one of super type $A$ and another of Cartan type $B_2$ with matrix $\left(\begin{smallmatrix} -1 & \zeta \\ 1 & \zeta \end{smallmatrix}\right)$; this algebra is nonzero, see \texttt{b2superb.log}.

\subsubsection{Type $\superd{m}{n}$, $N > 2$}\label{subsec:type-D-super}
The Weyl groupoid has objects with generalized Dynkin diagrams of six possible shapes.

For the diagram $\xymatrix@R-6pt{ {\bf A}_{\theta-2}(q;i_1,\dots, i_j) \ar @{-}[r]^{\hspace*{1.5cm}
 q^{-1}}  & 	\overset{q}{\underset{\ }{\circ}} \ar  @{-}[r]^{q^{-2}}  & \overset{q^2}{\underset{\ }{\circ}}}$, there are 3 cases:

\noindent $\diamondsuit N>4$.
Then $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_{\theta}$, with defining
relations
\begin{align*}
& & y_{iii\pm1}&=0, \quad i < \theta-1, q_{ii}\neq -1;  \\
	y_{\theta-1\theta-1\theta-2}&=0; &
	y_{ij} &= \lambda_{ij}, \quad i < j - 1;
	\\
	y_{\theta\theta\theta-1}&=\lambda_{\theta\theta\theta-1}; & [y_{(i-1i+1)},y_i]_c&=\nu_i, \quad q_{ii}=-1, 2\leq i\leq \theta-2;\\
	y_{\theta-1\theta-1\theta-1\theta}&=\lambda_{\theta-1\theta-1\theta-1\theta}; &
	y_i^2&=\mu_i, \quad q_{ii}=-1.
\end{align*}
If $\lambda_{\theta-1\theta-1\theta-1\theta} =\lambda_{\theta\theta\theta-1} =0$, then $\widetilde{\mathcal E}_{\bq}/ \langle y_{\theta-1\theta} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, one of super type $A$ and another of Cartan type $A_1$. If either $\lambda_{\theta-1\theta-1\theta-1\theta}\neq 0$ or $\lambda_{\theta\theta\theta-1}\neq 0$, then $\nu_{\theta-2}=0$ if $q_{\theta-2\theta-2}=-1$;
hence $\widetilde{\mathcal E}_{\bq}/ \langle y_{\theta-2\theta-1} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, one of super type $A$ and another of Cartan type $B_2$.

\noindent $\diamondsuit N=4$.
Then $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_{\theta}$, with defining
relations
\begin{align*}
& & y_{iii\pm1}&=\lambda_{iii\pm1}, \,  i < \theta-1, q_{ii}\neq -1;  \\
y_{\theta-1\theta-1\theta-2}&=\lambda_{\theta-1\theta-1\theta-2}; &
y_{ij} &= \lambda_{ij}, \,  i < j - 1; \\
y_{\theta-1\theta-1\theta-1\theta}&=0; &
[y_{(i-1i+1)},y_i]_c&=\nu_i, \,  q_{ii}=-1,  i\neq \theta-1;\\
[y_{(\theta-2\theta)},y_{\theta-1\theta}]_c&=0; &
y_i^2&=\mu_i, \,  q_{ii}=-1.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{\theta-1\theta} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, one of super type $A$ and another of Cartan type $A_1$.

\noindent $\diamondsuit N=3$.
Then $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_{\theta}$, with defining
relations
\begin{align*}
& & y_{iii\pm1}&=\lambda_{iii\pm1}, \, i < \theta-1, q_{ii}\neq -1;  \\
y_{\theta-1\theta-1\theta-2}&=\lambda_{\theta-1\theta-1\theta-2}; &
y_{ij} &= \lambda_{ij}, \, i < j - 1; \\
y_{\theta\theta\theta-1}&=0; &
[y_{(i-1i+1)},y_i]_c&=\nu_i, \, q_{ii}=-1, i\neq \theta-1;\\
y_i^2&=\mu_i, \, q_{ii}=-1; &
[[y_{(\theta-2\theta)}&,y_{\theta-1}]_c,y_{\theta-1}]_c=\lambda_{(\theta-2\theta)}.
\end{align*}
Here, $q_{\theta-2\theta-2}=q$ if $\lambda_{(\theta-2\theta)}\neq 0$; hence we work as for Cartan type $C_\theta$.

For the diagram $\xymatrix@R-6pt{ {\bf A}_{\theta-2}(q;i_1,\dots, i_j) \ar @{-}[r]^{\hspace*{1.5cm} q} &
\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{q^{-2}}  & \overset{q^2}{\underset{\ }{\circ}}}$, there are 2 cases:

\noindent $\diamondsuit N \neq 4$. Then $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_{\theta}$, with defining
relations
\begin{align*}
 y_i^2=\mu_i,& \, q_{ii}=-1; &
 y_{iii\pm1}&=0, \, i < \theta-1, q_{ii}\neq -1;  \\
 y_{ij} = \lambda_{ij}, &\, i < j - 1; &
 [[y_{\theta-2\theta-1},y_{(\theta-2\theta)}&]_c,y_{\theta-1}]_c =0, \, q_{\theta-2\theta-2}=-1; \\
 & & [[[y_{(\theta-3\theta)},y_{\theta-1}]_c, y_{\theta-2} &]_c,y_{\theta-1}]_c =0, \, q_{\theta-2\theta-2}\neq-1;\\
 y_{\theta\theta\theta-1}&=\lambda_{\theta\theta\theta-1}; &
[y_{(i-1i+1)},y_i]_c&=\nu_i, \, q_{ii}=-1,  i\neq \theta-1.
\end{align*}
If $\lambda_{\theta\theta\theta-1}=0$, then $\widetilde{\mathcal E}_{\bq}/ \langle y_{\theta-1\theta} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, one of super type $A$ and another of type $A_1$. If $\lambda_{\theta\theta\theta-1}\neq 0$, then either $q_{\theta-2\theta-2} \neq -1$, or else $q_{\theta-2\theta-2}=-1$, $\nu_{\theta-2}=0$; in both cases $\widetilde{\mathcal E}_{\bq}/ \langle y_{\theta-2\theta-1} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, one of super type $A_{\theta-2}$ and another of super type $A_2$.

\noindent $\diamondsuit N=4$.
Then $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_{\theta}$, with defining
relations
\begin{align*}
& y_{ij} = \lambda_{ij}, \, i < j - 1; \qquad y_{iii\pm1}=\lambda_{iii\pm1}, \, i < \theta-1, q_{ii}\neq -1;  \\
\quad &
[[y_{\theta-2\theta-1},y_{(\theta-2\theta)}]_c,y_{\theta-1}]_c =\lambda_{(\theta-2\theta)}, \, q_{\theta-2\theta-2}=-1; \\
& y_{\theta-1\theta}^2=\mu_{\theta-1\theta}; \qquad
[[[y_{(\theta-3\theta)},y_{\theta-1}]_c,y_{\theta-2}]_c,y_{\theta-1}]_c=0, \, q_{\theta-2\theta-2}\neq-1;\\
& y_i^2=\mu_i, \, q_{ii}=-1; \qquad
[y_{(i-1i+1)},y_i]_c=\nu_i, \, q_{ii}=-1,  i\neq \theta-1.
\end{align*}
First consider $q_{\theta-2\theta-2}\neq-1$. If $\lambda_{iii\pm1}\neq 0$, $i=\theta-2$, $\mu_{\theta-1\theta}\neq 0$,  then $\widetilde{\mathcal E}_{\bq}/ \langle y_{\theta-4\theta-3} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, one of super type $A_{\theta-4}$ and another with matrix $\left( \begin{smallmatrix}
-1 & q & -1 & 1 \\ -1 & q & -1 & \pm 1 \\ -1 & q & -1 & -1 \\ 1 & \pm 1 & 1 & -1  \end{smallmatrix} \right)$; this
 algebra is not zero, see \texttt{CDrk4a.log}. If $\mu_{\theta-1\theta}=0$, then
$\widetilde{\mathcal E}_{\bq}/ \langle y_{\theta-1\theta} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, one of super type $A_{\theta-1}$ and another of type $A_1$. If $\lambda_{\theta-2\theta-2\theta-1}=0$, then $\widetilde{\mathcal E}_{\bq}/ \langle y_{\theta-2\theta-1} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, one of super type $A_{\theta-2}$ and another of Cartan type $A_2$. If $\lambda_{\theta-2\theta-2\theta-3}=0$, then $\widetilde{\mathcal E}_{\bq}/ \langle y_{\theta-3\theta-2} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, one of super type $A_{\theta-3}$ and the other is a subalgebra of \texttt{CDrk4a.log}.

Now fix $q_{\theta-2\theta-2}=-1$. A similar analysis reduces to the case $\nu_{\theta-2}\neq 0$, $\lambda_{(\theta-2\theta)}\neq 0$; here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{\theta-4\theta-3} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, one of super type $A_{\theta-4}$ and another with matrix $\left( \begin{smallmatrix}
-1 & \overline{q}a^{-1} & a^2 & -a^{-4} \\ a & -1 & a^{-1} & a^3 \\ a^{-2} & qa & -1 & a^{-2} \\ -a^4 & a^{-3} & -a^2 & -1  \end{smallmatrix} \right)$, $a\neq 0$; this
algebra is not zero, see \texttt{CDrk4b.log}.

For the diagram $\xymatrix@R-6pt{
&   \overset{-1}{\circ} \ar  @{-}[d]_{q^{-1}}\ar  @{-}[dr]^{q^2} &
\\
{\bf A}_{\theta-3}(q;i_1,\dots, i_j) \ar @{-}[r]_{\hspace*{1.5cm} q^{-1}}  &  \overset{q}{\underset{\ }{\circ}} \ar  @{-}[r]_{q^{-1}}  & \overset{-1}{\underset{\ }{\circ}} } $ there are 2 cases:

\noindent $\diamondsuit N\neq 4$.
Then $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_{\theta}$, with defining relations
\begin{align*}
& & y_{iii\pm1}=0, \quad  i <& \theta-1, q_{ii}\neq -1; \\
y_{\theta-2\theta-2\theta}&=0;  &
[y_{(i-1i+1)},y_i]_c&=\nu_i, \quad q_{ii}=-1, i\leq \theta-3;\\
y_i^2&=\mu_i, \quad q_{ii}=-1; &
y_{ij}&=\lambda_{ij}, \quad i < j - 1, i\neq \theta-2;
\end{align*}
\vspace*{-0.7cm}
\begin{align*}
&y_{(\theta-2\theta)}+q_{\theta-1\theta}(1+q^{-1})[y_{\theta-2\theta},y_{\theta-1}]_c-
q_{\theta-2\theta-1}(1-q^2)y_{\theta-1}y_{\theta-2\theta}=0.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{\theta-2\theta}, y_{\theta-1\theta} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, one of super type $A_{\theta-1}$ and another of type $A_1$.

\noindent $\diamondsuit N=4$.
Then $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_{\theta}$, with defining
relations
\begin{align*}
y_i^2&=\mu_i, \, q_{ii}=-1; &
y_{ij} &= \lambda_{ij}, \,  i < j - 1, i\neq \theta-2;  \\
y_{\theta-2\theta-2\theta}&=\lambda_{\theta-2\theta-2\theta};&
y_{iii\pm1}&=\lambda_{iii\pm1}, \,   i < \theta-1, q_{ii}\neq -1; \\
y_{\theta-1\theta}^2&=\mu_{\theta-1\theta}; &
[y_{(i-1i+1)},y_i]_c&=\nu_i, \,  q_{ii}=-1, i\leq \theta-3;
\end{align*}
\vspace*{-0.7cm}
\begin{align*}
&y_{(\theta-2\theta)}+q_{\theta-1\theta}(1+q^{-1})[y_{\theta-2\theta},y_{\theta-1}]_c-
q_{\theta-2\theta-1}(1-q^2)y_{\theta-1}y_{\theta-2\theta}=0.
\end{align*}
Note that $\lambda_{\theta-2\theta-2\theta} \lambda_{\theta-2\theta-2\theta-1} =0$, so we may assume $\lambda_{\theta-2\theta-2\theta}=0$. If $\mu_{\theta-1\theta}=0$, then $\widetilde{\mathcal E}_{\bq}/ \langle y_{\theta-2\theta}, y_{\theta-1\theta} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, one of super type $A_{\theta-1}$ and another of type $A_1$. If $\mu_{\theta-1\theta} \neq 0$, $\lambda_{\theta-2\theta-2\theta-1}=0$, then $\widetilde{\mathcal E}_{\bq}/ \langle y_{\theta-2\theta}, y_{\theta-2\theta-1} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, one of super type $A_{\theta-2}$ and another of Cartan type $A_2$.
If $\mu_{\theta-1\theta}\lambda_{\theta-2\theta-2\theta-1} \neq 0$, $\lambda_{\theta-2\theta-2\theta-3}=0$, then $\widetilde{\mathcal E}_{\bq}/ \langle y_{\theta-3\theta-2} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, one of super type $A_{\theta-2}$ and a subalgebra of that in \texttt{CDrk4c.log}.
If $\mu_{\theta-1\theta} \lambda_{\theta-2\theta-2\theta-1} \lambda_{\theta-2\theta-2\theta-3} \neq 0$, then  $\nu_{\theta-3}=0$ and $\widetilde{\mathcal E}_{\bq}/ \langle y_{\theta-4\theta-3} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, one of super type $A_{\theta-4}$ and another with matrix $\left( \begin{smallmatrix}
-1 & q & -1 & -1 \\ -1 & q & -1 & \pm 1 \\ -1 & q & -1 & 1 \\ -1 & \pm q^{-1} & -1 & -1  \end{smallmatrix} \right)$; this
algebra is not zero, see \texttt{CDrk4c.log}.

For the diagram $\xymatrix@R-6pt{
&   \overset{-1}{\circ} \ar  @{-}[d]_{q^{-1}}\ar  @{-}[dr]^{q^2} &
\\
	{\bf A}_{\theta-3}(q;i_1,\dots, i_j) \ar @{-}[r]_{\hspace*{1.5cm} q}  &
	\overset{-1}{\underset{\
		}{\circ}} \ar  @{-}[r]_{q^{-1}}  & \overset{-1}{\underset{\ }{\circ}}
}$ there are 2 cases:

\noindent $\diamondsuit N\neq 4$.
Then $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_{\theta}$, with defining
relations
\begin{align*}
&[y_{\theta-3\theta-2\theta},y_{\theta-2}]_c=\nu_{\theta-2}';&
& y_{iii\pm1}=\lambda_{iii\pm1}, \quad  i <\theta-2, q_{ii}\neq -1; \\
&& &[y_{(i-1i+1)},y_i]_c=\nu_i, \quad q_{ii}=-1, i\leq \theta-3;\\
&y_i^2=\mu_i, \quad q_{ii}=-1 &
&y_{ij} = \lambda_{ij}, \quad i < j - 1, i\neq \theta-2;
\end{align*}
\vspace*{-0.7cm}
\begin{align*}
&y_{(\theta-2\theta)}+q_{\theta-1\theta}(1+q^{-1})[y_{\theta-2\theta},y_{\theta-1}]_c-
q_{\theta-2\theta-1}(1-q^2)y_{\theta-1}y_{\theta-2\theta}=0.
\end{align*}
Here, $\nu_{\theta-2}'\nu_{\theta-2}'=0$, we assume $\nu_{\theta-2}'=0$. Thus $\widetilde{\mathcal E}_{\bq}/ \langle y_{\theta-2\theta}, y_{\theta-1\theta} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, of super type $A_{\theta-1}$ and of type $A_1$.

\noindent $\diamondsuit N=4$.
Then $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_{\theta}$, with defining
relations
\begin{align*}
& y_{ij} = \lambda_{ij}, \quad i < j - 1, i\neq \theta-2; &
& y_{\theta-1 \theta}^2=\mu_{\theta-1 \theta} \\
& y_{iii\pm1}=\lambda_{iii\pm1}, \quad  i<\theta-2, q_{ii}\neq -1; &
& [y_{\theta-3\theta-2\theta},y_{\theta-2}]_c=\nu_{\theta-2}'; \\
&[y_{(i-1i+1)},y_i]_c=\nu_i, \quad q_{ii}=-1, i\leq \theta-3; &
& y_i^2=\nu_i, \quad q_{ii}=-1;
\end{align*}
\vspace*{-0.7cm}
\begin{align*}
&y_{(\theta-2\theta)}+q_{\theta-1\theta}(1+q^{-1})[y_{\theta-2\theta},y_{\theta-1}]_c-
2 q_{\theta-2\theta-1}y_{\theta-1}y_{\theta-2\theta}=0.
\end{align*}
Note that $\nu_{\theta-2} \lambda_{\theta-2}'=0$, so we may assume $\nu_{\theta-2}'=0$. If $\mu_{\theta-1\theta}=0$, then $\widetilde{\mathcal E}_{\bq}/ \langle y_{\theta-2\theta}, y_{\theta-1\theta} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, one of super type $A_{\theta-1}$ and another of type $A_1$. If $\mu_{\theta-1\theta} \neq 0$, $\nu_{\theta-2}=0$, then $\widetilde{\mathcal E}_{\bq}/ \langle y_{\theta-2\theta}, y_{\theta-2\theta-1} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, one of super type $A_{\theta-2}$ and another of Cartan type $A_2$.
If $\mu_{\theta-1\theta}\nu_{\theta-2} \neq 0$, then $q_{\theta-3\theta-3}=-1$ and $\nu_{\theta-3}=0$; hence $\widetilde{\mathcal E}_{\bq}/ \langle y_{\theta-4\theta-3}, y_{\theta-2\theta} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, one of super type $A_{\theta-4}$ and another of Cartan type $C_4$.

For the diagram $\xymatrix@R-6pt{
&   \overset{q}{\circ} \ar  @{-}[d]^{q^{-1}} &
\\
	{\bf A}_{\theta-3}(q;i_1,\dots, i_j) \ar @{-}[r]_{\hspace*{1.5cm} q^{-1}}  &
	\overset{q}{\underset{\
		}{\circ}} \ar  @{-}[r]_{q^{-1}}  & \overset{q}{\underset{\ }{\circ}}
}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_{\theta}$, with defining
relations
\begin{align*}
y_i^2=\nu_i, \quad & q_{ii}=-1 &
y_{iii\pm1}&=\lambda_{iii\pm1}, \quad  i < \theta-1, q_{ii}\neq -1; \\
y_{\theta-2\theta-2\theta}&=\lambda_{\theta-2\theta-2\theta}; &
y_{\theta-1\theta}&=0;\\
y_{\theta-1\theta-1\theta-2}&=\lambda_{\theta-1\theta-1\theta-2}; &
y_{ij} &= \lambda_{ij}, \quad i < j - 1, i\neq \theta-2; \\
y_{\theta\theta\theta-2}&=\lambda_{\theta\theta\theta-2}; &
[y_{(i-1i+1)},y_i]_c&=\nu_i, \,  q_{ii}=-1, i\leq \theta-3.
\end{align*}
If $N\neq 3$, then $\lambda_{\theta-2\theta-2\theta}= \lambda_{\theta\theta\theta-2}= \lambda_{\theta-1\theta-1\theta-2}= \lambda_{\theta-2\theta-2\theta-1}=0$; if $N=3$, then either $\lambda_{\theta-2\theta-2\theta}= \lambda_{\theta\theta\theta-2}=0$, or else $\lambda_{\theta-1\theta-1\theta-2}= \lambda_{\theta-2\theta-2\theta-1}=0$. Thus we may assume $\lambda_{\theta-2\theta-2\theta}= \lambda_{\theta\theta\theta-2}=0$, and
$\widetilde{\mathcal E}_{\bq}/ \langle y_{\theta-2\theta} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, one of super type $A_{\theta-1}$ and another of type $A_1$.

 For the diagram $\xymatrix@R-6pt{
&   \overset{q}{\circ} \ar  @{-}[d]^{q^{-1}} &
\\
{\bf A}_{\theta-3}(q;i_1,\dots, i_j) \ar @{-}[r]_{\hspace*{1.5cm} q}  &
\overset{-1}{\underset{\
}{\circ}} \ar  @{-}[r]_{q^{-1}}  & \overset{q}{\underset{\ }{\circ}} }$
the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_{\theta}$, with defining
relations
\begin{align*}
y_i^2=\mu_i, \,  & q_{ii}=-1;  &
y_{iii\pm1}&=\lambda_{iii\pm1}, \,   i < \theta-2, q_{ii}\neq -1; \\
[y_{\theta-3\theta-2\theta},y_{\theta-2}]_c&=\nu_{\theta-2}'; &
y_{\theta-1\theta}&=0;\\
y_{\theta-1\theta-1\theta-2}&=\lambda_{\theta-1\theta-1\theta-2}; &
y_{ij} &= \lambda_{ij}, \,  i < j - 1, i\neq \theta-2; \\
y_{\theta\theta\theta-2}&=\lambda_{\theta\theta\theta-2}; &
[y_{(i-1i+1)}&,y_i]_c=\nu_i, \,  q_{ii}=-1, i\leq \theta-2.
\end{align*}
As above, we may restrict to $\theta=4$ and $N=4$. As $\nu_{2}\nu_{2}'=0$, hence we may assume $\nu_{2}'=0$. Assume that $q_{11}=-1$, then $\nu_2=0$ and we can quotient onto a case of smaller rank, by making $y_{12}=0$. Otherwise, $q_{11}=q^{-1}$. If only $\nu_2\neq0$, we can project onto smaller ranks. Hence, we can restrict to the case $\nu_2=0$ as described in \S \ref{subsubsec:casebycase}. We can further project to smaller rank unless $\lambda_{112}\lambda_{332}\lambda_{442}\neq 0$, when ${\mathfrak q}=\left( \begin{smallmatrix}
q^{-1} & -1 & a^{-1}q & b^{-1} \\ q^{-1} & -1 & q & q \\ a & -1 & q & c^{-1} \\ b & -1 & c & q \end{smallmatrix} \right)$, for $a,b,c\in\{\pm q\}$. In any case, $\widetilde{\mathcal E}_{\bq}\neq 0$, see \texttt{CDrk4d.log}.

\subsubsection{Type $\superda{\alpha}$, $q,r,s\neq 1$, $qrs=1$}\label{subsec:type-D2-1-alpha}

The Weyl groupoid has four objects.

For $\xymatrix@C-4pt{\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{s} \ar@/^2pc/
@{-}[rr]^{q}
& \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{r}& \overset{-1}{\underset{\
}{\circ}}}$, the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_3$, with defining
relations
\begin{align*}
y_1^2&=\lambda_1; & y_2^2&=\lambda_2; & y_3^2&=\lambda_3;
\end{align*}
\vspace*{-0.7cm}
\begin{align*}
y_{(13)}-\frac{1-s}{q_{23}(1-r)}[y_{13},y_2]_c-q_{12}(1-s)y_2y_{13}=0.
\end{align*}
This algebra is nonzero by \cite[Lemma 5.16]{AAGMV}.

For $\xymatrix{ \overset{q}{\underset{\ }{\circ}}\ar  @{-}[r]^{q^{-1}}  &
\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{r^{-1}}  & \overset{r}{\underset{\
}{\circ}}}$, $q,r,s\neq -1$, the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_3$, with defining
relations
\begin{align*}
y_2^2&=\lambda_2; & y_{112}&=\lambda_1; & y_{332}&=\lambda_3; & y_{13}&=0,
\end{align*}
with $\lambda_1\lambda_3=0$. This algebra is nonzero, as it projects onto a nonzero algebra by making $y_1=0$ or $y_3=0$ according to $\lambda_1=0$ or $\lambda_3=0$.

For the same diagram and $q=-1,r,s\neq -1$, $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_3$, with defining
relations
\begin{align*}
y_1^2&=\lambda_1; & y_2^2&=\lambda_2; &
y_{12}^2&=\lambda_4; & y_{332}&=\lambda_3; & y_{13}&=0.
\end{align*}
Here $\lambda_3\neq0$ only if $r\in\mathrm{G}_4$. If $\lambda_3=0$, then this algebra is nonzero as it projects onto an algebra of Cartan type $A_2$.
Similarly, a projection to the rank-two case shows that $\widetilde{\mathcal E}_{\bq}\neq0$ if $\lambda_1=\lambda_2=\lambda_4=0$.

Assume $\lambda_3\neq0$ and pick $\epsilon\in\Bbbk$ such that $\epsilon^2=1$, i.e.~$\epsilon=\pm r^2$. Then:
 $\lambda_1\lambda_4\neq 0$ only if ${\mathfrak q}=\left(\begin{smallmatrix}
-1&1& \epsilon \\-1&-1&r\\\epsilon &r^2&r
\end{smallmatrix}\right)$.
 $\lambda_2\neq 0$ only if ${\mathfrak q}=\left(\begin{smallmatrix}
-1&\epsilon & x \\-\epsilon &-1&r\\ x^{-1} &r^2&r
\end{smallmatrix}\right)$, with $x^2=\epsilon$.
In any case, $\widetilde{\mathcal E}_{\bq}\neq0$, see \texttt{d21a.log}.

\medbreak

The remaining two generalized Dynkin diagrams are of the shape of of the last one, but with $s$
instead of $q$, respectively with $s$ instead of $r$.

\subsubsection{Type ${\mathbf F}(4)$, $N > 3$}\label{subsec:type-F-super}
The Weyl groupoid has objects with generalized Dynkin diagrams of six possible shapes.

	 For the diagram $\xymatrix{ \overset{\,\, q^2}{\underset{\ }{\circ}}\ar  @{-}[r]^{q^{-2}}  & \overset{\,\,
			q^2}{\underset{\ }{\circ}} \ar  @{-}[r]^{q^{-2}}  & \overset{q}{\underset{\ }{\circ}} & \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[l]_{q^{-1}}}$, there are 2 cases:

	\noindent $\diamondsuit N>4$.
	Then $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining
	relations
	\begin{align*}
	y_{13}&=0; & y_{14}&=0; & y_{24}&=0; \\
	y_{112}&=0; & y_{221}&=0; & y_{223}&=\lambda_{223};\\
	y_{334}&=0; & y_{3332}&=\lambda_{3332}; & y_{4}^2&=\mu_4.
	\end{align*}
	Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{34} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, one of type $B_{3}$ and another of type $A_1$.

	\noindent $\diamondsuit N=4$.
	Then $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining
	relations
	\begin{align*}
	y_{13}&=0; & y_{14}&=\lambda_{14}; & y_{24}&=\lambda_{24}; \\
	y_{334}&=\lambda_{334}; & [y_{(13)},y_2]_c&=0; & [y_{23},y_{(24)}]_c&=\lambda_{(24)}; \\
	y_{3332}&=0; & y_{4}^2&=\mu_4; & y_{12}^2&=\mu_{12}.
	\end{align*}
If $\lambda_{(24)}=0$, then $\widetilde{\mathcal E}_{\bq}/ \langle y_{23} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, one of type $A_2$ and another of super type $A_2$.
If $\lambda_{(24)}\neq0$ but all the other $\lambda$'s are 0, then $\widetilde{\mathcal E}_{\bq}/ \langle y_{12} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of type super $C_3$ and $A_1$.  The general case follows by applying Lemma
\ref{lem:nonzero-k<l}.

	 For the diagram $\xymatrix{ \overset{\,\, q^2}{\underset{\ }{\circ}}\ar  @{-}[r]^{q^{-2}}  & \overset{\,\,
			q^2}{\underset{\ }{\circ}} \ar  @{-}[r]^{q^{-2}}  & \overset{-1}{\underset{\ }{\circ}} &
		\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[l]_{q}}$, there are 2 cases:

	\noindent $\diamondsuit N>4$.
	Then $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining
	relations
	\begin{align*}
	y_{13}&=0; & y_{112}&=\lambda_{112}; & y_{24}&=0;  & y_{3}^2&=\mu_3; \\
	y_{14}&=0; & y_{221}&=\lambda_{221}; & y_{223}&=0; & y_{4}^2&=\mu_4; &
	[[y_{43},y_{432}]_c,y_3]_c&=0.
	\end{align*}
	Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{23} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, one of type $A_2$ and another of super type $A_2$.

	\noindent $\diamondsuit N=4$.
	Then $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining
	relations
	\begin{align*}
	y_{13}&=\lambda_{13}; & y_{14}&=\lambda_{14}; &
	[y_{(13)},y_2]_c&=\nu_2; &  y_{3}^2&=\mu_3;  \\
	y_{23}^2&=\mu_{23}; & y_{24}&=\lambda_{24}; &
	[[y_{43},y_{432}]_c,y_3]_c&=\lambda_{432}; & y_{4}^2&=\mu_4.
	\end{align*}
	If $\nu_{2}=0$, then $\widetilde{\mathcal E}_{\bq}/ \langle y_{12} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, one of type $A_1$ and another of super type $CD$. If $\nu_2\neq 0$, then $\lambda_{432}=0$ and $\widetilde{\mathcal E}_{\bq}/ \langle y_{34} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, of types $A_3$ and $A_1$.

	 For the diagram $\xymatrix{ \overset{\,\, q^2}{\underset{\ }{\circ}}\ar  @{-}[r]^{q^{-2}}  & \overset{\,\,
			q^2}{\underset{\ }{\circ}} \ar  @{-}[r]^{q^{-2}}  & \overset{-1}{\underset{\ }{\circ}}\ar @{-}[r]^{q^3} & \overset{q^{-3}}{\underset{\ }{\circ}}}$, there are 3 cases:

	\noindent $\diamondsuit N\neq 4,6$.
	Then $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining
	relations
	\begin{align*}
	y_{443}&=\lambda_{443}; & y_{3}^2&=\mu_3; & y_{13}&=\lambda_{13}; & y_{14}&=0; \,  y_{24}=0; \\
	y_{112}&=0; & y_{221}&=0; & y_{223}&=\lambda_{223}; &
	[[[y_{432},y_3]_c,&[y_{4321},y_3]_c]_c,y_{32}]_c=0.
	\end{align*}
	Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{12} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, one of type $A_1$ and another of super type $D(2,1;\alpha)$.

	\noindent $\diamondsuit N=6$.
	Then $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining
	relations
	\begin{align*}
	y_{3}^2&=\mu_2; & y_{34}^2&=0;  & y_{13}&=0; & y_{14}&=0; \quad y_{24}=0; \\
	y_{112}&=\lambda_{112}; & y_{221}&=\lambda_{221}; & y_{223}&=0; &
	[[[ y_{432} & ,y_3]_c,[y_{4321},y_3]_c]_c,y_{32}]_c=0.
	\end{align*}
	Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{23} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of type $A_2$.

	\noindent $\diamondsuit N=4$.
	Then $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining
	relations
	\begin{align*}
	y_{3}^2&=\mu_3;  & y_{13}&=\lambda_{13}; & y_{14}&=0; & [y_{(13)},y_2]_c&=\nu_2; \\
	y_{443}&=\lambda_{443}; &  y_{23}^2&=\mu_{23}; & y_{24}&=0; &
	[[[y_{432},y_3]_c,&[y_{4321},y_3]_c]_c,y_{32}]_c=0.
	\end{align*}
	If $\nu_{2}=0$, then $\widetilde{\mathcal E}_{\bq}/ \langle y_{12} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, one of type $A_1$ and another of super type $CD$. If $\nu_2\neq 0$, then $\lambda_{443}=0$ and $\widetilde{\mathcal E}_{\bq}/ \langle y_{34} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, of types $A_3$ and
	$A_1$.

	 For the diagram $\xymatrix{ \overset{\,\, q^2}{\underset{\ }{\circ}}\ar  @{-}[r]^{q^{-2}}  &
		\overset{q}{\underset{\ }{\circ}} \ar  @{-}[r]^{q^{-1}}  & \overset{-1}{\underset{\ }{\circ}}\ar  @{-}[r]^{q^3} & \overset{q^{-3}}{\underset{\ }{\circ}}}$, there are 3 cases:

	\noindent $\diamondsuit N\neq 4,6$.
	Then $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining
	relations
	\begin{align*}
	y_{3}^2&=\mu_3; &  y_{13}&=0; & y_{14}&=0; & y_{24}&=0; \\
	y_{443}&=\lambda_{443}; &  y_{112}&=\lambda_{221}; & y_{2221}&=\lambda_{2221}; & y_{223}&=0;
	\end{align*}
	\vspace*{-0.7cm}
	\begin{align*}
	[[y_{(14)},y_2]_c,y_3]_c-q_{23}(q^2-q)[[y_{(14)},y_3]_c,y_2]_c=0.
	\end{align*}
	Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{23} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, one of type $B_2$ and another of super type $A_2$.

	\noindent $\diamondsuit N=6$.
	Then $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining
	relations
	\begin{align*}
	y_{3}^2&=\mu_3; & y_{13}&=0; & y_{14}&=0; & y_{24}&=0; \\
	y_{34}^2&=\mu_{34}; &  y_{112}&=0; & y_{2221}&=0; & y_{223}&=0;
	\end{align*}
	\vspace*{-0.7cm}
	\begin{align*}
	[[y_{(14)},y_2]_c,y_3]_c-q_{23}(q^2-q)[[y_{(14)},y_3]_c,y_2]_c=0.
	\end{align*}
	Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{23} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, one of type $B_2$ and another of super type $A_2$.

	\noindent $\diamondsuit N=4$.
	Then $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining
	relations
	\begin{align*}
	y_{3}^2&=\mu_3; &  y_{13}&=\lambda_{13}; & y_{14}&=0; & y_{24}&=0; \\
	y_{443}&=\lambda_{443}; &  [y_{12},y_{(13)}]_c&=\lambda_{(13)}; & y_{2221}&=0; & y_{223}&=\lambda_{223};
	\end{align*}
	\vspace*{-0.7cm}
	\begin{align*}
	[[y_{(14)},y_2]_c,y_3]_c-q_{23}(q^2-q)[[y_{(14)},y_3]_c,y_2]_c=0.
	\end{align*}
If $\lambda_{(13)}=0$, then $\widetilde{\mathcal E}_{\bq}/ \langle y_{12} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, one of type $A_1$ and another of type $D(2,1;\alpha)$.
If $\lambda_{(13)}\neq0$ but all the other $\lambda$'s are 0, then $\widetilde{\mathcal E}_{\bq}/ \langle y_{34} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $B_3$ and $A_1$. The general case follows by applying Lemma \ref{lem:nonzero-k<l}.

	 For the diagram $\xymatrix@R-6pt@C-4pt{ & & \overset{q}{\circ}\ar  @{-}[ld]_{q^{-1}} \ar
		@{-}[d]^{q^{-1}}
		\\ \overset{q^2}{\underset{\ }{\circ}} \ar  @{-}[r]^{q^{-2}}  &  \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{q^{2}}  & \overset{-1}{\underset{\ }{\circ}}}$ there are 2 cases:

	\noindent $\diamondsuit N>4$.
	Then $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining
	relations
	\begin{align*}
	y_{2}^2&=\mu_2; &  y_{13}&=0; & y_{14}&=0; & y_{112}&=\lambda_{112}; \\
	y_{3}^2&=\mu_3; &  y_{442}&=0; & y_{443}&=0; & [y_{(13)},y_2]_c&=0;
	\end{align*}
	\vspace*{-0.7cm}
	\begin{align*}
	y_{(24)}- q_{34}q[y_{24},y_3]_c-q_{23}(1-q^{-1})y_3y_{24}=0.
	\end{align*}
	Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{23},y_{24} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of super type $A_2$.

	\noindent $\diamondsuit N=4$.
	Then $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining
	relations
	\begin{align*}
	y_{2}^2&=\mu_2; &  y_{13}&=\lambda_{13}; & y_{14}&=0; & y_{12}^2&=\mu_{12}; \\
	y_{3}^2&=\mu_3; &  y_{442}&=\lambda_{442}; & y_{443}&=\lambda_{443}; & [y_{(13)},y_2]_c&=\nu_2;
	\end{align*}
	\vspace*{-0.7cm}
	\begin{align*}
	y_{(24)}- q_{34}q[y_{24},y_3]_c-q_{23}(1-q^{-1})y_3y_{24}=0.
	\end{align*}
	If $\lambda_{442}\neq 0$, then $\nu_{2}=\lambda_{443}=0$, so $\widetilde{\mathcal E}_{\bq}/ \langle y_{23}, y_{34} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, of types $A_1$ and $C_3$.
	If $\lambda_{443}\neq 0$, then $\nu_{2}=\lambda_{442}=0$, so $\widetilde{\mathcal E}_{\bq}/ \langle y_{23}, y_{24} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, of types $A_2$ and super $A_2$.
	If $\nu_{2}\neq 0$, then $\lambda_{442}=\lambda_{443}=0$, so $\widetilde{\mathcal E}_{\bq}/ \langle y_{24}, y_{34} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, of types $A_1$ and $A_3$.

	 For the diagram $\xymatrix@R-6pt@C-4pt{ & & \overset{-1}{\circ}\ar  @{-}[ld]_{q^{2}} \ar  @{-}[d]^{q^{-3}}
		\\ \overset{q^2}{\underset{\ }{\circ}} \ar  @{-}[r]^{q^{-2}}  &  \overset{-1}{\underset{\
			}{\circ}} \ar  @{-}[r]^{q}  & \overset{-1}{\underset{\ }{\circ}}}$ there are 2 cases:

	\noindent $\diamondsuit N>4$.
	Then $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining
	relations
	\begin{align*}
	y_{2}^2&=\mu_2; & y_{13}&=0; & y_{14}&=0; &   [y_{124},y_2]_c&=0; \\
	y_{3}^2&=\mu_3; &   y_{112}&=\lambda_{112}; & y_{4}^2&=\mu_4; &  [[y_{32},y_{321}]_c,y_2]_c&=0;
	\end{align*}
	\vspace*{-0.7cm}
	\begin{align*}
	y_{(24)}+ q_{34}\frac{1-q^3}{1-q^2}[y_{24},y_3]_c-q_{23}(1-q^{-3})y_3y_{24}=0.
	\end{align*}
	Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{23},y_{24} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of super type $A_2$.

	\noindent $\diamondsuit N=4$.
	Then $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining
	relations
	\begin{align*}
	y_{2}^2&=\mu_2; & y_{13}&=\lambda_{13}; & y_{14}&=\lambda_{14}; &   [y_{124},y_2]_c&=\nu_2; \\
	y_{3}^2&=\mu_3; &   y_{12}^2&=\mu_{12}; & y_{4}^2&=\mu_4; &  [[y_{32},y_{321}]_c,y_2]_c&=\lambda_{(321)};
	\end{align*}
	\vspace*{-0.7cm}
	\begin{align*}
	y_{(24)}+ q_{34}\frac{1-q^3}{1-q^2}[y_{24},y_3]_c-q_{23}(1-q^{-3})y_3y_{24}=0.
	\end{align*}
	If $\lambda_{(321)}=0$, then $\widetilde{\mathcal E}_{\bq}/ \langle y_{23}, y_{34} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, of types $A_1$ and $A_3$.
	If $\lambda_{(321)}\neq 0$, then $\nu_{2}=0$, so  $\widetilde{\mathcal E}_{\bq}/ \langle y_{24}, y_{34} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, of types $A_1$ and $A_3$.

\subsubsection{Type  ${\mathbf G}(3)$, $N > 3$}\label{subsec:type-G-super}
The Weyl groupoid has four objects.

For $\xymatrix{ \overset{-1}{\underset{\ }{\circ}}\ar  @{-}[r]^{q^{-1}}  &
\overset{q}{\underset{\ }{\circ}} \ar  @{-}[r]^{q^{-3}}  & \overset{\,\, q^3}{\underset{\
}{\circ}}}$, $N\neq 4,6$, the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_3$, with defining
relations
\begin{align*}
y_{13}&=0; & y_{221}&=0; &  y_{332}&=0; &
 y_{22223}&=\lambda_2; & y_{1}^2&=\lambda_1,
\end{align*}
with $\lambda_1\lambda_2=0$. If $\lambda_1\neq 0$, then $\widetilde{\mathcal E}_{\bq}$ projects onto $\Bbbk\langle y_1|y_1^2=\lambda_1\rangle$. If $\lambda_2\neq0$, then $\widetilde{\mathcal E}_{\bq}$ projects onto a nonzero algebra related to Cartan case $G_2$.

When $N =6$, $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_3$, with defining
relations
\begin{align*}
y_{13}&=\lambda_{3}; & y_{221}&=0; &  [y_{12},y_{(13)}]_c&=0; &
 y_{22223}&=0; & y_{1}^2&=\lambda_1.
\end{align*}
Now $\widetilde{\mathcal E}_{\bq}\neq 0$ by projecting onto the case $y_2=0$.

For $N =4$, $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_3$, with defining
relations
\begin{align*}
y_{13}&=0; & y_{221}&=\lambda_2; &  [[[y_{(13)},&y_2]_c,y_2]_c,y_2]_c=0; &
 y_{332}&=0; & y_{1}^2&=\lambda_1.
\end{align*}
The longest relation is undeformed by \texttt{g3aN4.log}. Here $\widetilde{\mathcal E}_{\bq}$ projects onto a nonzero algebra related to Cartan case $A_2$.

\medskip

For $\xymatrix{ \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{q}  &
\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{q^{-3}}  & \overset{\,\, q^3}{\underset{\
}{\circ}}}$, $N\neq 6$, the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_3$, with defining
relations
\begin{align*}
y_{13}&=0; & y_{332}&=\lambda_3; &  [[y_{12},&[y_{12},y_{(13)}]_c]_c,y_2]_c=0; &
 y_{1}^2&=\lambda_1; & y_{2}^2&=\lambda_2.
\end{align*}
The third relation is undeformed by \texttt{g3b.log}. Here $\lambda_3\neq 0$ only when $N=12$. In any case $\widetilde{\mathcal E}_{\bq}\neq0$ by making $y_{12}=0$ as in \S \ref{subsubsec:cut}.

When $N=6$, $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_3$, with defining
relations
\begin{align*}
y_{13}&=\lambda_3; & y_{23}^2&=\lambda_5; &  [[y_{12},&[y_{12},y_{(13)}]_c]_c,y_2]_c=\lambda_4; &
 y_{1}^2&=\lambda_1; & y_{2}^2&=\lambda_2.
\end{align*}
If $\lambda_4=0$, then we can project onto a case of smaller rank again making $y_{12}=0$.
If $\lambda_4\neq 0$, then ${\mathfrak q}=\left(\begin{smallmatrix}
-1      & a  & -a^{-4}\\
a^{-1}q &-1  & -a^3\\
-a^4&  a^{-3}  &-1
\end{smallmatrix}\right)$. Here, $\widetilde{\mathcal E}_{\bq}\neq 0$ by \texttt{g3b2.log}.

For $\xymatrix{ \overset{-q^{-1}}{\underset{\ }{\circ}} \ar  @{-}[r]^{q^2}  &
\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{q^{-3}}  & \overset{\,\, q^3}{\underset{\
}{\circ}}}$, $N\neq 6$, the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_3$, with defining
relations
\begin{align*}
y_{13}&=\lambda_3; \qquad y_{332}=\lambda_4; \qquad y_{1112}=0; \qquad y_{2}^2=\lambda_2; \\
[y_1, & [y_{123},y_2]_c]_c =\frac{q_{12}q_{32}}{1+q}[y_{12},y_{123}]_c-q^{-1}(1-q^{-1})q_{12}q_{13} y_{123}y_{12}.
\end{align*}
Here $\lambda_3\neq 0$ only if $N=4$ and $\lambda_4\neq0$ only if $N=12$; also $\lambda_2\lambda_3=0$. Hence, we can project onto a case of smaller rank and  $\widetilde{\mathcal E}_{\bq}\neq0$. The longest relation is undeformed by \texttt{g3c.log}.

When $N =6$, $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_3$, with defining
relations
\begin{align*}
y_{13}&=0; \qquad y_{23}^2=\lambda_3; \qquad [y_{112},y_{12}]_c=\lambda_1; \qquad y_{2}^2=\lambda_2;  \\
[y_1, & [y_{123},y_2]_c]_c = \frac{q_{12}q_{32}}{1+q}[y_{12},y_{123}]_c-q^{-1}(1-q^{-1})q_{12}q_{13} y_{123}y_{12}.
\end{align*}
We see that $\widetilde{\mathcal E}_{\bq}\neq 0$ using the arguments in \S \ref{subsubsec:casebycase}.  The longest relation is undeformed by \texttt{g3c.log}.

\medskip

For $\xymatrix@C-4pt{\overset{q}{\underset{\ }{\circ}} \ar  @{-}[r]^{q^{-2}} \ar@/^2pc/
@{-}[rr]^{q^{-1}}
& \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{q^{3}}& \overset{-1}{\underset{\
}{\circ}}}$, the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_3$, with defining
relations
\begin{align*}
y_{1112}&=\lambda_1; & y_{2}^2&=\lambda_2; & y_3^2&=\lambda_3; &
 y_{113}&=\lambda_4;
\end{align*}
\vspace*{-0.6cm}
\begin{align*}
y_{(13)}+q^{-2}q_{23}\frac{1-q^3}{1-q}[y_{13},y_2]_c-q_{12}(1-q^3)y_2y_{13}=0.
\end{align*}
If $\lambda_1=\lambda_4=0$, then $\widetilde{\mathcal E}_{\bq}\neq 0$ by \cite[Lemma 5.16]{AAGMV}. Now, $\lambda_1\neq0$ only if $N=6$ and $\lambda_4\neq0$ only if $N=4$.
If $\lambda_1\neq 0$ and $\lambda_3=0$ or if $\lambda_4\neq0$ and $\lambda_2=0$ we may project onto a case of smaller rank. Indeed, $\lambda_2\lambda_4\neq 0$ and we can have
$\lambda_1\lambda_3\neq 0$ only if
${\mathfrak q}
=\left(\begin{smallmatrix}
q&-1&\pm1\\ q&-1&\pm1\\ \pm q^{-1}& \pm q^{-1} & -1
\end{smallmatrix}\right)$. In this case, $\widetilde{\mathcal E}_{\bq}\neq0$ by \texttt{g3d.log}.

\subsection{Modular type, characteristic
3}\label{sec:by-diagram-modular-char3}

\subsubsection{Type $\mathfrak{br}(2, a)$, $\zeta \in \mathrm{G}_3$, $q\notin \mathrm{G}_3$}\label{subsec:type-br(2,a)}
The Weyl groupoid has two objects: $\xymatrix{ \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{q^{-1}}  &
\overset{q}{\underset{\ }{\circ}}}$ and $\xymatrix{ \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta^2q}  &
\overset{\zeta q^{-1}}{\underset{\ }{\circ}}}$. Notice that the right hand diagram is of the shape of the left hand one, with $\zeta q^{-1}$
instead of $q$. Therefore, we only have to analyze one of them: we concentrate on the later.

If $q\neq -1$, then the algebra  $\widetilde{\mathcal E}_{\bq}$ is generated by $y_1, y_{2}$ with defining relations
\begin{align*}
y_{1}^3 &= \lambda_1; &  y_{221} &=\lambda_2,
\end{align*}
with $\lambda_1\lambda_2=0$. Also, $\lambda_2=0$ unless $q=-\zeta$ and ${\mathfrak q}=\left(\begin{smallmatrix}
\zeta& -\zeta \\\zeta& -\zeta
\end{smallmatrix}\right)$.
This algebra is nonzero: If $\lambda_1\neq 0$, this is \cite[Lemma 5.16]{AAGMV}. When $\lambda_2\neq0$, see \texttt{br2a-q.log}.

If $q=-1$, then the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_1, y_{2}$
with defining relations
\begin{align*}
y_{1}^3 &= \lambda_1; & y_{2}^{2} &=\lambda_2; & [y_{112}, y_{12}]_c&=\lambda_3,
\end{align*}
with, $\lambda_3=0$ unless ${\mathfrak q}=\left(\begin{smallmatrix}
\zeta& -1 \\ 1& -1
\end{smallmatrix}\right)$.
This algebra is nonzero, see \texttt{br2a-1.log}.

\subsubsection{Type $\mathfrak{br}(3)$, $\zeta \in \mathrm{G}'_{9}$}\label{subsec:type-br(3)}
The Weyl groupoid has two objects.

For $\xymatrix{ \overset{\zeta}{\underset{\ }{\circ}}\ar  @{-}[r]^{\overline{\zeta}}  &
\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\overline{\zeta}^{\,
3}}{\underset{\ }{\circ}}}$, the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_3$, with defining
relations
\begin{align*}
y_{13}&=0; & y_{112}&=0; & y_3^3&=\lambda_1; & [[y_{332},& y_{3321}]_c,y_{32}]_c=0; &
 y_{221}&=0; & y_{223}&=0.
\end{align*}
This algebra is nonzero by \cite[Lemma 5.16]{AAGMV}.

For $\xymatrix{ \overset{\zeta}{\underset{\ }{\circ}}\ar  @{-}[r]^{\overline{\zeta}}  &
\overset{\overline{\zeta}^{\, 4}}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta^{4}}  & \overset{\overline{\zeta}^{\,
3}}{\underset{\ }{\circ}}}$, the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_3$, with defining
relations
\begin{align*}
y_{13}&=0; & y_{112}&=0; & y_3^3&=\lambda_1; &
 y_{2221}&=0; & y_{223}&=0;
\end{align*}
\vspace*{-0.7cm}
\begin{align*}
 [[y_{(13)}, y_{2}]_c,y_{3}]_c-(1+\zeta^4)^{-1}q_{23}[[y_{(13)}, y_{3}]_c,y_{2}]_c=0.
\end{align*}
This algebra is nonzero by \cite[Lemma 5.16]{AAGMV}.

\subsection{Super modular type, characteristic
3}\label{sec:by-diagram-super-modular-char3}

\subsubsection{\ Type $\mathfrak{brj}(2; 3)$, $\zeta \in \mathrm{G}'_9$}\label{subsec:type-brj(2,3)}
The Weyl groupoid has three objects.

For $\xymatrix{ \overset{-\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}^{\, 2}}  &
\overset{\zeta^3}{\underset{\ }{\circ}}}$, $\widetilde{\mathcal E}_{\bq}$ is generated by $y_1, y_2$ with defining relations
\begin{align*}
y_2^3&=\lambda_2; &  [y_1,[y_{12},y_2]_c]_c+\frac{\zeta^5(1-\zeta)q_{12}}{1-\zeta^7}
y_{12}^2&=0;&
 y_{1112}&=0.
\end{align*}
This algebra is nonzero as it projects onto $\Bbbk\langle y_2|y_2^3=\lambda_2\rangle$.

For $\xymatrix{ \overset{\zeta^3}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  &
\overset{-1}{\underset{\ }{\circ}}}$, $\widetilde{\mathcal E}_{\bq}$ is generated by $y_1, y_2$ with defining relations
\begin{align*}
y_1^{3}&=\lambda_1; & y_2^2&=\lambda_2; &  [y_{112},[y_{112},y_{12}]_c]_c&=0,
\end{align*}
with $\lambda_1\lambda_2=0$. The last relation is undeformed, see \texttt{brj23.log}.
Hence  this algebra is nonzero by \cite[Lemma 5.16]{AAGMV}.

For $\xymatrix{ \overset{-\zeta^2}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  &
\overset{-1}{\underset{\ }{\circ}}}$, $\widetilde{\mathcal E}_{\bq}$ is generated by $y_1, y_2$ with defining relations
\begin{align*}
y_2^2&=\lambda_2; &  [y_{112},y_{12}]_c&=0; &
 y_{111112}&=0.
\end{align*}
This algebra is nonzero by \cite[Lemma 5.16]{AAGMV}.

\subsubsection{\ Type $\mathfrak{g}(1,6)$, $\zeta \in \mathrm{G}'_{3} \cup \mathrm{G}'_{6}$}\label{subsec:type-g(1,6)}
The Weyl groupoid has two objects.

	 For $\xymatrix{ \overset{\zeta}{\underset{\ }{\circ}}\ar  @{-}[r]^{\overline{\zeta}}  &
		\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}^{\, 2}}  &
		\overset{-1}{\underset{\ }{\circ}}}$, $\zeta \in \mathrm{G}'_{6}$, the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_3$, with defining relations
	\begin{align*}
	y_{13}&=0; &y_{112}&=0; &  y_{221}&=0; &
	y_{2223}&=\lambda_{2223}; &y_{3}^2&=\mu_3.
	\end{align*}
	Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{12} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of super type $A_2$ and $A_1$.

	 For $\xymatrix{ \overset{\zeta}{\underset{\ }{\circ}}\ar  @{-}[r]^{\overline{\zeta}}  &
		\overset{-\overline{\zeta}}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta^{2}}  & \overset{-1}{\underset{\
			}{\circ}}}$, $\zeta \in \mathrm{G}'_{6}$, the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_3$, with defining relations
	\begin{align*}
	y_{13}&=0; & y_{112}&=;  & y_{3}^2&=\mu_3; & [y_{223},y_{23}]_c&=\lambda_{23};
	\end{align*}
	\vspace*{-0.7cm}
	\begin{align*}
	[y_2, [y_{21},y_{23}]_c]_c+q_{13}q_{23}q_{21}[y_{223},y_{21}]_c+q_{21}y_{21}y_{223}=0.
	\end{align*}
	Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{12} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of super type $B_2$ and $A_1$.

	 For $\xymatrix{ \overset{\zeta}{\underset{\ }{\circ}}\ar  @{-}[r]^{\overline{\zeta}}  &
		\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}^{\, 2}}  &
		\overset{-1}{\underset{\ }{\circ}}}$, $\zeta \in \mathrm{G}'_{3}$, the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_3$, with defining relations
	\begin{align*}
	y_{13}&=0; & y_{112}&=\lambda_{112}; &  [[y_{(13)},&y_2]_c,y_2]_c=0;\\
	y_{3}^2&=\mu_3; & y_{221}&=\lambda_{221}; & [y_{223},y_{23}]_c&=\lambda_{23}.
	\end{align*}
	If $\lambda_{23}=0$, then $\widetilde{\mathcal E}_{\bq}/ \langle y_{23} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, of types $A_2$ and $A_1$.
	If $\lambda_{23}\neq 0$, then $\lambda_{112}=\lambda_{221}=0$, so  $\widetilde{\mathcal E}_{\bq}/ \langle y_{12} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, of types $A_1$ and super $B_2$.

	 For $\xymatrix{ \overset{\zeta}{\underset{\ }{\circ}}\ar  @{-}[r]^{\overline{\zeta}}  &
		\overset{-\overline{\zeta}}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta^{2}}  & \overset{-1}{\underset{\
			}{\circ}}}$, $\zeta \in \mathrm{G}'_{3}$, the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_3$, with defining relations
	\begin{align*}
	y_{13}&=0; &y_{112}&=0; &  y_{2221}&=0; &
	y_{2223}&=\lambda_{2223}; &y_{3}^2&=\mu_3;
	\end{align*}
	\vspace*{-0.7cm}
	\begin{align*}
	[y_{1},y_{223}]_c+q_{23}[y_{(13)},y_{2}]_c+(\zeta^2-\zeta)q_{12}y_{2}y_{(13)}=0.
	\end{align*}
	Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{12} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of super type $B_2$ and $A_1$.

\subsubsection{\ Type $\mathfrak{g}(2,3)$, $\zeta \in \mathrm{G}'_{3}$}\label{subsec:type-g(2,3)}
The Weyl groupoid has four objects.

	 For $\xymatrix{ \overset{-1}{\underset{\ }{\circ}}\ar  @{-}[r]^{\overline{\zeta}}  &
		\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\
			}{\circ}}}$, the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_3$, with defining relations
	\begin{align*}
	y_{13}&=\lambda_{13}; & y_{221}&=0; & y_1^2&=\mu_1; &  y_3^2&=\mu_3;
	\end{align*}
	\vspace*{-0.7cm}
	\begin{align*}
	[y_{223},y_{23}]_c&=\lambda_{23}; &
	[[y_{(13)},y_2]_c,y_2]_c&=\lambda_{(13)}.
	\end{align*}
	If $\lambda_{(13)}=0$, then $\widetilde{\mathcal E}_{\bq}/ \langle y_{12} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, of types $A_1$ and super $B_2$.
	If $\lambda_{(13)}\neq 0$, then ${\mathfrak q}=\left( \begin{smallmatrix}
	-1 & b & -b^{-3} \\ \zeta^2b^{-1} & \zeta & \zeta b \\ -b^3 & b^{-1} & -1 \end{smallmatrix} \right)$, $b\neq 0$; this
	algebra is not zero, see \texttt{g23-1.log}.

	 For $\xymatrix{ \overset{-1}{\underset{\ }{\circ}}\ar  @{-}[r]^{\zeta}  &
		\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\
			}{\circ}}}$, the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_3$, with defining relations
	\begin{align*}
	y_{13}&=\lambda_{13}; & y_{1}^2&=\mu_1; & y_2^2&=\mu_2; &  y_3^2&=\mu_3;
	\end{align*}
	\vspace*{-0.7cm}
	\begin{align*}
	[[y_{12},y_{(13)}]_c,y_2]_c&=\lambda_{123}; & [[y_{32},y_{321}]_c,y_2]_c&=\lambda_{321}.
	\end{align*}
	If $\lambda_{123}=\lambda_{321}=0$, then $\widetilde{\mathcal E}_{\bq}/ \langle y_{12} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, of types $A_1$ and super $A_2$.
	If $\lambda_{123}\neq 0$ (similar for $\lambda_{321}\neq 0$), then ${\mathfrak q}=\left( \begin{smallmatrix}
	-1 & b & b^{-3} \\ \zeta b^{-1} & -1 & -\zeta b^2 \\ b^3 & -b^{-2} & -1 \end{smallmatrix} \right)$, $b\neq 0$; this
	algebra is not zero, see \texttt{g23-2.log}.  Observe that $\lambda_{123}\lambda_{321}\neq 0$ only if $b^3=-1$.

	 For $\xymatrix{ \overset{-1}{\underset{\ }{\circ}}\ar  @{-}[r]^{\overline{\zeta}}  &
		\overset{-\overline{\zeta}}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\
			}{\circ}}}$, the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_3$, with defining relations
	\begin{align*}
	y_{13}&=\lambda_{13}; & y_{2221}&=\lambda_{2221}; & y_{2223}&=\lambda_{2223}; &  y_1^2&=\mu_1; & y_3^2&=\mu_3;
	\end{align*}
	\vspace*{-0.7cm}
	\begin{align*}
	[y_{1},y_{223}]_c+q_{23}[y_{(13)},y_2]_c-(1-\zeta)q_{12}y_2y_{(13)}=0.
	\end{align*}
	As $\lambda_{2221}\lambda_{2223}=0$, we may assume $\lambda_{2221}=0$, and $\widetilde{\mathcal E}_{\bq}/ \langle y_{12} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, of types super $B_2$ and $A_1$.

	 For $\xymatrix{ \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}} \ar@/^2pc/
		@{-}[rr]^{\overline{\zeta}}  & \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}
		& \overset{\zeta}{\underset{\ }{\circ}}  }$, the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_3$, with defining relations
	\begin{align*}
	y_{112}&=0; & y_{113}&=0; & y_{331}&=0; &  y_{332}&=0; & y_2^2&=\mu_2;
	\end{align*}
	\vspace*{-0.7cm}
	\begin{align*}
	y_{(13)} -
	q_{23}\zeta [y_{13},y_{2}]_c-q_{12}(1-\overline{\zeta})y_2y_{13}=0.
	\end{align*}
	Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{12},y_{23} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, of types $A_2$ and $A_1$.

\subsubsection{\ Type $\mathfrak{g}(3, 3)$, $\zeta \in \mathrm{G}'_3$}\label{subsec:type-g(3,3)}
The Weyl groupoid has six objects.

	 For $\xymatrix{ \overset{\overline{\zeta}}{\underset{\ }{\circ}}\ar  @{-}[r]^{\zeta}  &
		\overset{\overline{\zeta}}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{\zeta}{\underset{\
			}{\circ}}
		\ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\ }{\circ}}}$
	the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining relations
	\begin{align*}
	y_{223}&=0; & y_{13}&=\lambda_{13}; &y_{14}&=0; & y_{112}&=\lambda_{112}; & [y_{3321},&y_{32}]_c=\lambda_{321};\\
	y_{334}&=0; &  y_{4}^2&=\mu_4 & y_{24}&=0; & y_{221}&=\lambda_{221}; & [[y_{(24)},&y_{3}]_c,y_3]_c=0.
	\end{align*}
	Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{34} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, of types $B_3$ and $A_1$.

	 For $ \xymatrix{ \overset{\overline{\zeta}}{\underset{\ }{\circ}}\ar  @{-}[r]^{\zeta}  &
		\overset{\overline{\zeta}}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\
			}{\circ}}
		\ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\ }{\circ}}}$
	the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining relations
	\begin{align*}
	y_{13}&=0; & y_{3}^2&=\mu_3; &y_{14}&=0; & y_{112}&=\lambda_{112}; & [[y_{43},&y_{432}]_c,y_3]_c=0;\\
	& & y_{4}^2&=0; & y_{24}&=0; & y_{221}&=\lambda_{221}; & y_{223}&=0.
	\end{align*}
	Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{34} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, of types super $A_3$ and $A_1$.

	 For $\xymatrix@R-6pt{  &    \overset{-1}{\circ} \ar  @{-}[d]^{\overline{\zeta}} & \\
		\overset{\overline{\zeta}}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\
			}{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\ }{\circ}}}$
	the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining relations
	\begin{align*}
	y_{13}&=0; & y_{3}^2&=\mu_3; &y_{14}&=0; & y_{34}&=\lambda_{34}; & [y_{124},&y_{2}]_c=0;\\
	&  & y_{4}^2&=\mu_4; & y_{112}&=0; & y_{2}^2&=\mu_2; & [y_{324},&y_{2}]_c=\nu_2.
	\end{align*}
	Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{12} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, of types super $A_3$ and $A_1$.

	 For $\xymatrix@R-6pt{  &    \overset{-1}{\circ} \ar  @{-}[d]^{\zeta} & \\
		\overset{\overline{\zeta}}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{\overline{\zeta}}{\underset{\
			}{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\ }{\circ}}}$
	the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining relations
	\begin{align*}
	y_{13}&=0; & y_{3}^2&=\mu_3; &y_{14}&=0; & y_{34}&=\lambda_{34}; & y_{112}&=\lambda_{112};\\
	&& y_{4}^2&=\mu_4; & y_{221}&=\lambda_{221}; & y_{223}&=0; & y_{224}&=0.
	\end{align*}
	Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{34} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, of types super $A_3$ and $A_1$.

	 For $\xymatrix@R-6pt{  &    \overset{-1}{\circ} \ar  @{-}[d]^{\overline{\zeta}} & \\
		\overset{\overline{\zeta}}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{\zeta}{\underset{\
			}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\ }{\circ}}}$
	the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining relations
	\begin{align*}
	y_{13}&=0; &  y_{3}^2&=\mu_3; & y_{14}&=0; & y_{34}&=\lambda_{34}; & [[y_{(13)},&y_2]_c,y_2]_c=0;\\
	y_{112}&=0; &  y_{4}^2&=\mu_4; & y_{223}&=0; & y_{224}&=0; & [[y_{124},&y_2]_c,y_2]_c=0.
	\end{align*}
	Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{12} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, of types super $A_3$ and $A_1$.

	 For $\xymatrix@R-6pt{  &    \overset{-1}{\circ} \ar  @{-}[d]_{\overline{\zeta}}\ar  @{-}[dr]^{\overline{\zeta}} & \\
		\overset{\overline{\zeta}}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\
			}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}}}$
	the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining relations
	\begin{align*}
	y_{13}&=\lambda_{13}; & y_{2}^2&=\mu_2; & y_{14}&=0; & y_{112}&=0; & [y_{(13)},&y_{2}]_c=\nu_2;\\
	&  & y_{4}^2&=\mu_4; & y_{332}&=0; & y_{334}&=0; & [y_{124},&y_{2}]_c=0;
	\end{align*}
	\vspace*{-0.7cm}
	\begin{align*}
	y_{(24)}-\zeta q_{34}[y_{24},y_3]_c-q_{23}(1-\overline{\zeta})y_3y_{24}=0.
	\end{align*}
	Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{24}, y_{34} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, of types super $A_3$ and $A_1$.

\subsubsection{\ Type $\mathfrak{g}(4,3)$, $\zeta \in \mathrm{G}'_3$}\label{subsec:type-g(4,3)}
The Weyl groupoid has ten objects.

	 For $\xymatrix{ \overset{\overline{\zeta}}{\underset{\ }{\circ}}\ar  @{-}[r]^{\zeta}  &
		\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
			}{\circ}}
		\ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\ }{\circ}}}$,
	the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining relations
	\begin{align*}
	y_{13}&=\lambda_{13};  & y_{2}^2&=\mu_2; &y_{14}&=0; & y_{24}&=\lambda_{14}; & [[y_{(24)},&y_3]_c,y_3]_c=\lambda_{(24)};\\
	[y_{(13)}&,y_2]_c=\nu_2;  & y_{4}^2&=\mu_4; & y_{112}&=0; & y_{332}&=0; & [y_{334},&y_{34}]_c=\lambda_{34}.
	\end{align*}
	If $\nu_2=0$, then $\widetilde{\mathcal E}_{\bq}/ \langle y_{12} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, of types super $\mathfrak{g}(2,3)$ and $A_1$. If $\nu_2\neq 0$, then $\lambda_{(24)}=\lambda_{34}=0$, so $\widetilde{\mathcal E}_{\bq}/ \langle y_{34} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, of types super $A_3$ and $A_1$.

	 For $\xymatrix{ \overset{\overline{\zeta}}{\underset{\ }{\circ}}\ar  @{-}[r]^{\zeta}  &
		\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-\overline{\zeta}}{\underset{\
			}{\circ}}
		\ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\ }{\circ}}}$,
	the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining relations
	\begin{align*}
	y_{13}&=0; & y_{4}^2&=\mu_4; &y_{14}&=0; & y_{24}&=\lambda_{24}; & [y_{(13)},y_2]_c&=0;\\
	&  & y_{2}^2&=\mu_2;  & y_{112}&=0; & y_{3332}&=\lambda_{3332}; & y_{3334}&=\lambda_{3334};
	\end{align*}
	\vspace*{-0.7cm}
	\begin{align*}
	[y_2,y_{334}]_c-q_{34}[y_{(24)},y_3]_c+(\zeta^2-\zeta)q_{23}y_3y_{(24)}=0.
	\end{align*}
	Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{12} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, of types super $\mathfrak{g}(2,3)$ and $A_1$.

	 For $\xymatrix{ \overset{-1}{\underset{\ }{\circ}}\ar  @{-}[r]^{\overline{\zeta}}  &
		\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\
			}{\circ}}
		\ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\ }{\circ}}}$,
	the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining relations
\begin{align*}
y_{1}^2&=\mu_1; & y_{2}^2&=\mu_2; & y_{13}&=\lambda_{13};  & [y_{(13)}&,y_2]_c=\nu_2; \\
& & y_{3}^2&=\mu_3; &y_{14}&=\lambda_{14}; & [[y_{23},&y_{(24)}]_c,y_3]_c=\lambda_{234};\\
& & y_{4}^2&=\mu_4; & y_{24}&=\lambda_{24}; &  [[y_{43},&y_{432}]_c,y_3]_c=\lambda_{432}.
\end{align*}
If $\nu_2=0$, then $\widetilde{\mathcal E}_{\bq}/ \langle y_{12} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, of types super $\mathfrak{g}(2,3)$ and $A_1$. If $\nu_2\neq 0$, then $\lambda_{234}=\lambda_{432}=0$, so $\widetilde{\mathcal E}_{\bq}/ \langle y_{34} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, of types super $A_3$ and $A_1$.

	 For $\xymatrix{ \overset{-1}{\underset{\ }{\circ}}\ar  @{-}[r]^{\zeta}  &
		\overset{\overline{\zeta}}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\
			}{\circ}}
		\ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\ }{\circ}}}$,
	the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining relations
	\begin{align*}
	y_{13}&=\lambda_{13}; & y_{3}^2&=\mu_3; &y_{14}&=\lambda_{14}; & y_{24}&=0; & [[y_{43},&y_{432}]_c,y_3]_c=0;\\
	&  & y_{4}^2&=\mu_4; & y_{221}&=0; & y_{223}&=0; & y_1^2&=\mu_2.
	\end{align*}
	This algebra is nonzero by \cite[Lemma 5.16]{AAGMV}.

 For $\xymatrix{ \overset{-1}{\underset{\ }{\circ}}\ar  @{-}[r]^{\overline{\zeta}}  &
		\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{\zeta}{\underset{\
			}{\circ}}
		\ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\ }{\circ}}}$,
the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining relations
\begin{align*}
[y_{(13)},y_2]_c&=0; & y_{1}^2&=\mu_1; & y_{13}&=0; &  [[y_{(24)},y_{3}]_c,y_3]_c&=\lambda_{(24)}; \\
y_{334}&=0; & y_{2}^2&=\mu_2; & y_{14}&=\lambda_{14}; & [y_{3321},y_{32}]_c&=0; \\
& & y_{4}^2&=\mu_4; & y_{24}&=\lambda_{24}; & [y_{332},y_{32}]_c&=\lambda_{32}.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{12} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, of types super $\mathfrak{g}(2,3)$ and $A_1$.

	 For $\xymatrix{ \overset{-1}{\underset{\ }{\circ}}\ar  @{-}[r]^{\zeta}  &
		\overset{\overline{\zeta}}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{\zeta}{\underset{\
			}{\circ}}
		\ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\ }{\circ}}}$,
	the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining relations
	\begin{align*}
	y_{13}&=0;  & y_{1}^2&=\mu_1; &y_{14}&=\lambda_{14}; & y_{24}&=0; & [[y_{(24)},&y_{3}]_c,y_3]_c=0;\\
	y_{334}&=0;  & y_{4}^2&=\mu_4; & y_{221}&=0; & y_{223}&=0; & [y_{3321},&y_{32}]_c=0.
	\end{align*}
	This algebra is nonzero by \cite[Lemma 5.16]{AAGMV}.

	 For $\xymatrix@R-6pt{  &    \overset{\overline{\zeta}}{\circ} \ar  @{-}[d]^{\zeta} & \\
		\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\
			}{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\ }{\circ}}}$,
	the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining relations
	\begin{align*}
	y_{13}&=0;  & y_2^2&=\mu_2; &y_{14}&=\lambda_{14}; & y_{34}&=0; & [y_{124},&y_2]_c=\nu_2';\\
	[y_{(13)},y_2]_c&=0;  & y_3^2&=\mu_3; &  y_{112}&=0; & y_{442}&=0; & [[y_{32},&y_{324}]_c,y_2]_c=0.
	\end{align*}
	Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{12} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, of types super $CD$ and $A_1$.

	 For $\xymatrix@R-6pt{  &    \overset{-1}{\circ} \ar  @{-}[d]_{\overline{\zeta}}\ar  @{-}[dr]^{\overline{\zeta}} &
		\\
		\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
			}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}}}$,
	the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining relations
	\begin{align*}
	y_{13}&=0; &y_{14}&=\lambda_{14}; & y_{221}&=0; & y_{223}&=\lambda_{223}; & y_{224}&=0;\\
	y_{332}&=\lambda_{332}; & y_{334}&=0;  & y_1^2&=\mu_1; & y_4^2&=\mu_4;
	\end{align*}
	\vspace*{-0.7cm}
	\begin{align*}
	y_{(24)}-\zeta q_{34}[y_{24},y_3]_c-(1-\overline{\zeta})q_{23}y_3y_{24}=0.
	\end{align*}
	Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{24}, y_{34} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, of types super $A_3$ and $A_1$.

	 For $\xymatrix@R-6pt{  &    \overset{\overline{\zeta}}{\circ} \ar  @{-}[d]^{\zeta} & \\
		\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
			}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\ }{\circ}}}$,
	the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining relations
	\begin{align*}
	y_{13}&=0;  & y_{442}&=0; &y_{14}&=\lambda_{14}; & y_{34}&=0; & [[y_{124},&y_2]_c,y_2]_c=\lambda_{124}; \\
	y_3^2&=0; & y_{223}&=0; & y_{112}&=\lambda_{112}; & y_{221}&=\lambda_{221}; & [[y_{324},&y_2]_c,y_2]_c=0.
	\end{align*}
	Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{23} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, of types $C_3$ and $A_1$.

	 For $\xymatrix@R-6pt{  &    \overset{-1}{\circ} \ar  @{-}[d]_{\overline{\zeta}}\ar  @{-}[dr]^{\overline{\zeta}} & \\
		\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\
			}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}}}$,
	the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining relations
	\begin{align*}
	y_{13}&=0; & y_2^2&=\mu_2; & y_{14}&=\lambda_{14}; & y_{332}&=0; & y_{334}&=0; \\
	&  & y_4^2&=\mu_4; & [y_{(13)},&y_2]_c=0; & y_{1}^2&=\mu_1; & [y_{124},&y_2]_c=\nu_2';
	\end{align*}
	\vspace*{-0.7cm}
	\begin{align*}
	y_{(24)}-\zeta q_{34}[y_{24},y_3]_c-(1-\overline{\zeta})q_{23}y_3y_{24}=0.
	\end{align*}
	Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{23}, y_{34} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, of types super $A_3$ and $A_1$.

\subsubsection{\ Type $\mathfrak{g}(3, 6)$, $\zeta \in \mathrm{G}'_3$}\label{subsec:type-g(3,6)}
The Weyl groupoid has seven objects.

	 For $\xymatrix{ \overset{-1}{\underset{\ }{\circ}}\ar  @{-}[r]^{\overline{\zeta}}  &
		\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
			}{\circ}}\ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\ }{\circ}}}$,
	the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining relations
	\begin{align*}
	y_{13}&=0; & y_{4}^2&=\mu_4; &y_{14}&=\lambda_{14}; & y_{24}&=0; & [[y_{(24)},&y_{3}]_c,y_{3}]_c=0;\\
	y_{332}&=\lambda_{332}; &y_{1}^2&=0;  & y_{221}&=0; & y_{223}&=\lambda_{223}; & [y_{334},&y_{34}]_c=\lambda_{34}.
	\end{align*}
	Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{12} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, of types super $\mathfrak{g}(1,6)$ and $A_1$.

	 For $\xymatrix{ \overset{-1}{\underset{\ }{\circ}}\ar  @{-}[r]^{\overline{\zeta}}  &
		\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-\overline{\zeta}}{\underset{\
			}{\circ}}
		\ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\ }{\circ}}}$,
	the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining relations
	\begin{align*}
	y_{13}&=0;  & y_{1}^2&=\mu_1;  &y_{14}&=\lambda_{14}; & y_{24}&=0; & y_{221}&=0;\\
	& & y_{4}^2&=\mu_4; & y_{223}&=0; & y_{3332}&=0; & y_{3334}&=\lambda_{3334};
	\end{align*}
	\vspace*{-0.7cm}
	\begin{align*}
	[y_2,y_{334}]_c+q_{34}[y_{(24)},y_3]_c+(\zeta^2-\zeta)q_{23}y_3y_{(24)}=0.
	\end{align*}
	Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{12} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, of types super $\mathfrak{g}(1,6)$ and $A_1$.

For $\xymatrix{ \overset{-1}{\underset{\ }{\circ}}\ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\ }{\circ}}}$,
	the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining relations
	\begin{align*}
	y_1^2&=\mu_1; & y_{13}&=0; &y_{14}&=\lambda_{14}; & y_{24}&=\lambda_{24}; & [[y_{(24)},&y_{3}]_c,y_{3}]_c=\lambda_{(24)};\\
	y_{2}^2&=\mu_2; & y_{4}^2&=\mu_4; & y_{332}&=0; & [y_{(13)},& y_{2}]_c=0; & [y_{334},&y_{34}]_c=\lambda_{34}.
	\end{align*}
	Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{12} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, of types $\mathfrak{g}(2,3)$ and $A_1$.

	 For $\xymatrix{ \overset{-1}{\underset{\ }{\circ}}\ar  @{-}[r]^{\zeta}  &
		\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-\overline{\zeta}}{\underset{\
			}{\circ}}
		\ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\ }{\circ}}}$,
	the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining relations
	\begin{align*}
	y_{13}&=0;  & y_{2}^2&=\mu_2; &y_{14}&=\lambda_{14}; & y_{24}&=\lambda_{24}; & [y_{(13)},&y_2]_c=0;\\
	&  & y_{4}^2&=\mu_4; & y_{3332}&=\lambda_{3332}; & y_{3334}&=\lambda_{3334}; & y_{1}^2&=\mu_1;
	\end{align*}
	\vspace*{-0.7cm}
	\begin{align*}
	[y_2,y_{334}]_c+q_{34}[y_{(24)},y_3]_c+(\zeta^2-\zeta)q_{23}y_3y_{(24)}=0.
	\end{align*}
	Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{12} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, of types super $\mathfrak{g}(2,3)$ and $A_1$.

 For $\xymatrix{ \overset{\zeta}{\underset{\ }{\circ}}\ar  @{-}[r]^{\overline{\zeta}}  &
		\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{\zeta}{\underset{\
			}{\circ}}
		\ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\ }{\circ}}}$,
the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining relations
\begin{align*}
y_{2}^2&=\mu_2; & y_{13}&=0; & y_{112}&=0; & [y_{332},&y_{32}]_c=\lambda_{32}; \\
y_{4}^2&=\mu_4; & y_{14}&=0; & y_{334}&=0; & [[y_{(24)},&y_3]_c,y_3]_c=\lambda_{(24)};\\
& & y_{24}&=\lambda_{24}; & [y_{(13)},&y_2]_c=0; & [y_{3321},&y_{32}]_c=0.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{12} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, of types super $\mathfrak{g}(2,3)$ and $A_1$.

	 For $\xymatrix{ \overset{\zeta}{\underset{\ }{\circ}}\ar  @{-}[r]^{\overline{\zeta}}  &
		\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\
			}{\circ}}
		\ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\ }{\circ}}}$,
	the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining relations
	\begin{align*}
	y_{13}&=0;  & y_{3}^2&=\mu_3; & [y_{(13)},&y_{2}]_c=0; & y_{24}&=\lambda_{24}; & [[y_{23},&y_{(24)}]_c,y_{3}]_c=\lambda_{234};\\
	y_{14}&=0;  & y_{4}^2&=\mu_4; & y_{112}&=0; & y_{2}^2&=\mu_2; & [[y_{43},&y_{432}]_c,y_{3}]_c=\lambda_{432}.
	\end{align*}
	Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{12} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components, of types super $\mathfrak{g}(2,3)$ and $A_1$.

For $\xymatrix@R-6pt{  &    \overset{-1}{\circ} \ar  @{-}[d]_{\overline{\zeta}}\ar  @{-}[dr]^{\overline{\zeta}} & \\  \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}}}$,
	the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining relations
	\begin{align*}
	y_{13}&=0; &y_{14}&=0; & y_{221}&=\lambda_{221}; & y_{223}&=\lambda_{223}; & y_{224}&=0;\\
	y_{332}&=\lambda_{332}; & y_{334}&=0;  & y_{112}&=\lambda_{112}; & y_4^2&=\mu_4;
	\end{align*}
	\vspace*{-0.7cm}
	\begin{align*}
	y_{(24)}-\zeta q_{34}[y_{24},y_3]_c-(1-\overline{\zeta})q_{23}y_3y_{24}=0.
	\end{align*}
	Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{24}, y_{34} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $A_3$ and $A_1$.

\subsubsection{\ Type $\mathfrak{g}(2, 6)$, $\zeta \in \mathrm{G}'_3$}\label{subsec:type-g(2,6)}
The Weyl groupoid has four objects.

 For $\xymatrix{\overset{\zeta}{\underset{\ }{\circ}}\ar  @{-}[r]^{\overline{\zeta}}  &
	\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\
		}{\circ}}
	\ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  &
	\overset{\zeta}{\underset{\ }{\circ}}}$
the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
\begin{align*}
& & y_{112}&=\lambda_{112}; & y_{223}&=0; & [[[y_{(14)},y_3]_c , y_2]_c , y_3]_c &=0; \\
& & y_{3}^2&=\mu_3; & y_{554}&=\lambda_{554}; &
[[[y_{5432},y_3]_c, y_4]_c , y_3]_c &=0; \\
y_{221}&=\lambda_{221}; & y_{443}&=0; &
y_{445}& =\lambda_{225}; & y_{ij}=\lambda_{ij}, \, i<j, \, & \widetilde{q}_{ij}=1.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{23},\rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $A_3$ and $A_2$.

 For $\xymatrix@R-8pt{  &    \overset{-1}{\circ} \ar  @{-}[d]_{\zeta}\ar  @{-}[dr]^{\zeta} &
	\\
	\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\
		}{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\ }{\circ}}\ar  @{-}[r]^{\overline{\zeta}}  &
	\overset{\zeta}{\underset{\ }{\circ}}}$
the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
\begin{align*}
y_{112}&=0; & y_{3}^2&=\mu_3; & y_{ij}&=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;  & [y_{125},& y_2]_c=0;\\
y_{443}&=0; & y_{5}^2&=\mu_5; & [y_{(13)},& y_2]_c=0; & [y_{(24)},& y_3]_c=0;
\end{align*}
\vspace*{-0.7cm}
\begin{align*}
y_{2}^2&=\mu_2; & [y_{435},& y_3]_c=0; & y_{235}-q_{35}\overline{\zeta} [y_{25},y_3]_c-q_{23}(1-\zeta)y_3y_{25}=0.
\end{align*}
This algebra is nonzero by \cite[Lemma 5.16]{AAGMV}.

 For $\xymatrix@R-10pt{ & &    \overset{\zeta}{\circ} \ar  @{-}[d]^{\overline{\zeta}} & \\
	\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}} & \overset{\zeta}{\underset{\
		}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}
	& \overset{-1}{\underset{\ }{\circ}}}$
the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
\begin{align*}
y_{223}&=0; & y_{112}&=\lambda_{112}; & y_{221}&=\lambda_{221}; & y_{3}^2&=\mu_3; \quad
[y_{435}, y_3]_c=0;\\
 [y_{(24)},& y_3]_c=0; & y_{4}^2&=\mu_4; & y_{553}&=0; & y_{ij}&=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{23},\rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $A_3$ and $A_2$.

 For $\xymatrix@R-10pt{  &  &  \overset{\zeta}{\circ} \ar  @{-}[d]^{\overline{\zeta}} & \\
	\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}} & \overset{\zeta}{\underset{\
		}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}
	& \overset{-1}{\underset{\ }{\circ}}}$
the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
\begin{align*}
y_{4}^2&=\mu_4; \qquad y_{553}=\lambda_{553}; & y_{112}&=\lambda_{112}; & y_{332}&=\lambda_{332}; & y_{221}&=\lambda_{221};\\
y_{ij}&=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;  & y_{335}&=\lambda_{335}; & y_{223}&= \lambda_{223}; & y_{334}&=0.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{34},\rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $A_4$ and $A_1$.

\subsubsection{\ Type $\mathfrak{el}(5;3)$, $\zeta \in \mathrm{G}'_3$}\label{subsec:type-el(5;3)}
The Weyl groupoid has fifteen objects.

 For $\xymatrix{\overset{\zeta}{\underset{\ }{\circ}}\ar  @{-}[r]^{\overline{\zeta}}  &
	\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
		}{\circ}}
	\ar  @{-}[r]^{\overline{\zeta}}  & \overset{\overline{\zeta}}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  &
	\overset{-1}{\underset{\ }{\circ}}}$
the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
\begin{align*}
y_{221}&=\lambda_{221}; & y_{5}^2&=\mu_5; & y_{334}&=0; & [[y_{(35)},&y_4]_c,y_4]_c=0;\\
& & y_{445}&=0; & y_{112}&=\lambda_{112}; & y_{ij}&=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1; \\
& & y_{223}&=\lambda_{223}; & y_{332}&=\lambda_{332}; & [y_{4432},&y_{43}]_c=\lambda_{432}.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{45},\rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $B_4$ and $A_1$.

 For $\xymatrix{\overset{\zeta}{\underset{\ }{\circ}}\ar  @{-}[r]^{\overline{\zeta}}  &
	\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
		}{\circ}}
	\ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  &
	\overset{-1}{\underset{\ }{\circ}}}$
the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
\begin{align*}
& & y_{4}^2&=\mu_4; & y_{112}&=\lambda_{112}; & y_{221}&=\lambda_{221}; &  &[[y_{54},y_{543}]_c,y_4]_c=0;\\
y_{334}&=0; & y_{5}^2&=\mu_5; & y_{223}&=\lambda_{223}; & y_{332}&=\lambda_{332}; & &y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{45},\rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $A_4$ and $A_1$.

 For $\xymatrix{\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta} &
	\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\
		}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}
	& \overset{\zeta}{\underset{\ }{\circ}}}$
the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
\begin{align*}
y_{443}&=0;      &  y_{445}&=\lambda_{445}; & [[y_{23},&y_{(24)}]_c,y_3]_c=0; & y_{1}^2&=\mu_1;\quad y_{2}^2=\mu_2\\
y_{3}^2&=\mu_3;  & y_{554}&=\lambda_{554};  & [y_{(13)},&y_2]_c=\nu_2;         & y_{ij}&=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{34},\rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $A_3$ and $A_2$.

 For $\xymatrix{\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}} &
	\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{\overline{\zeta}}{\underset{\
		}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}
	& \overset{\zeta}{\underset{\ }{\circ}}}$
the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations $y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;$
\begin{align*}
y_{2}^2&=\mu_2; & y_{112}&=0; & y_{445}&=\lambda_{445}; & [[y_{(24)},&y_{3}]_c,y_3]_c=0; \\
y_{332}&=0; & y_{443}&=0; & y_{554}&=\lambda_{554}; & [y_{3345},&y_{34}]_c=\lambda_{345};& [y_{(13)},&y_2]_c=\nu_2.
\end{align*}
If $\nu_2=0$, then $\widetilde{\mathcal E}_{\bq}/ \langle y_{23},\rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $B_3$ and $A_2$.
If $\nu_2\neq 0$, then $\lambda_{345}=0$, so $\widetilde{\mathcal E}_{\bq}/ \langle y_{34},\rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $A_3$ and $A_2$.

 For $\xymatrix@C-4pt{\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}} &
	\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\
		}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}
	& \overset{\zeta}{\underset{\ }{\circ}}}$
the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
$y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;$
\begin{align*}
& & y_{3}^2&=\mu_3; & y_{445}&=\lambda_{445}; & y_{221}&=0; & [[[y_{5432},&y_{3}]_c,y_4]_c,y_3]_c=0;
\\
y_{223}&=0; & y_{1}^2&=\mu_1; & y_{554}&=\lambda_{554}; & y_{443}&=0; & [[[y_{(14)},&y_{3}]_c,y_2]_c,y_3]_c=\lambda_{(14)}.
\end{align*}
If $\lambda_{(14)}=0$, then $\widetilde{\mathcal E}_{\bq}/ \langle y_{34}\rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $A_3$, $A_2$. If $\lambda_{(14)}\neq 0$ but all the other $\lambda$'s are zero, then $\widetilde{\mathcal E}_{\bq}/ \langle y_{45}\rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $B_4$, $A_1$. The general case follows by Lemma \ref{lem:nonzero-k<l}.

 For $\xymatrix@R-8pt{  &  &  \overset{-1}{\circ} \ar  @{-}[dl]_{\zeta}\ar  @{-}[d]^{\zeta} &
	\\
	\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\
		}{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}
	& \overset{\zeta}{\underset{\ }{\circ}}}$
the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
$y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;$
\begin{align*}
y_{2}^2&=\mu_2; & [y_{125},&y_{2}]_c=\nu_2'; & y_{3}^2&=\mu_3; & [y_{(24)},&y_{3}]_c=0;\\
[y_{(13)},&y_{2}]_c=\nu_2; & y_{5}^2&=\mu_5; &y_{1}^2&=\mu_1; & [y_{435},&y_{3}]_c=0;
\end{align*}
\vspace*{-0.7cm}
\begin{align*}
 y_{443}&=0; & y_{235}-q_{35}\overline{\zeta}[y_{25},y_3]_c-q_{23}(1-\zeta)y_3y_{25}=0.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{34}\rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types super $CD$ and $A_1$.

 For $\xymatrix@R-8pt{  &  & & \overset{-1}{\circ} \ar  @{-}[d]^{\zeta}\ar  @{-}[dl]_{\zeta}
	\\
	\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
		}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}
	& \overset{\overline{\zeta}}{\underset{\ }{\circ}}}$
the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
$y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;$
\begin{align*}
y_{3}^2&=\mu_3; & [y_{(24)},&y_{3}]_c=\nu_3; & y_{445}&=0; & y_{112}&=\lambda_{112};\\
y_{221}&=\lambda_{221}; & y_{223}&=0; & y_{443}&=0; & [y_{235},&y_{3}]_c=0;
\end{align*}
\vspace*{-0.7cm}
\begin{align*}
y_{5}^2&=\mu_5; & y_{(35)}-q_{45}\overline{\zeta}[y_{35},y_4]_c-q_{34}(1-\zeta)y_4y_{35}=0.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{35}, y_{45} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types super $A_4$ and $A_1$.

 For $\xymatrix@R-8pt{  &  &  \overset{-1}{\circ} \ar  @{-}[dl]_{\zeta}\ar  @{-}[d]^{\zeta} &
	\\
	\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{\overline{\zeta}}{\underset{\
		}{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}
	& \overset{\zeta}{\underset{\ }{\circ}}}$
the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
$y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;$
\begin{align*}
y_{1}^2&=\mu_1; & [y_{(24)},&y_{3}]_c=\nu_2; & y_{3}^2&=\mu_3; & y_{221}&=0;\\
y_{223}&=0; & y_{5}^2&=\mu_5; & y_{443}&=0; & [y_{435},&y_{3}]_c=0;
\end{align*}
\vspace*{-0.7cm}
\begin{align*}
 y_{225}&=0; & y_{235}-q_{35}\overline{\zeta}[y_{25},y_3]_c-q_{23}(1-\zeta)y_3y_{25}=0.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{25}, y_{35} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types super $A_4$ and $A_1$.

 For $\xymatrix@R-8pt{  &  &  \overset{\zeta}{\circ} \ar  @{-}[d]^{\overline{\zeta}} & \\
	\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\
		}{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{\overline{\zeta}}{\underset{\ }{\circ}} \ar
	@{-}[r]^{\zeta}  & \overset{-1}{\underset{\ }{\circ}}}$
the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
$y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;$
\begin{align*}
y_{112}&=0; & y_{2}^2&=\mu_2; & [y_{(13)},&y_2]_c=\nu_2; &
y_{4}^2&=\mu_4; & [[y_{235},&y_{3}]_c,y_3]_c=0;\\
& & y_{553}&=0; & y_{332}&=0; & y_{334}&=0; & [[y_{435},&y_{3}]_c,y_3]_c=0.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{35} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types super $A_4$ and $A_1$.

 For $\xymatrix@R-8pt{  &  &  \overset{\zeta}{\circ} \ar  @{-}[d]^{\overline{\zeta}} & \\
	\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\
		}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}
	& \overset{-1}{\underset{\ }{\circ}}}$
the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations $y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;$
\begin{align*}
& & y_{2}^2&=\mu_2; & [[y_{23},&y_{235}]_c,y_3]_c=0; &
y_{3}^2&=\mu_3; & [y_{(13)},&y_2]_c=\nu_2;\\
y_{4}^2&=\mu_4; & y_{553}&=0; & [y_{(24)},&y_3]_c=\nu_3; & y_{1}^2&=\mu_1; & [y_{435},&y_{3}]_c=0.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{35} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types super $A_4$ and $A_1$.

 For $\xymatrix@R-8pt{  &  &  \overset{\zeta}{\circ} \ar  @{-}[d]^{\overline{\zeta}} & \\
	\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
		}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}
	& \overset{-1}{\underset{\ }{\circ}}}$
the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations $y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;$
\begin{align*}
y_{1}^2&=\mu_1; & y_{3}^2&=\mu_3; & y_{223}&=0; & [[[y_{1235},&y_{3}]_c,y_2]_c,y_3]_c=\lambda_{1235};\\
y_{221}&=0; & [y_{(24)},&y_3]_c=0; &
y_{4}^2&=\mu_4; & y_{553}&=0; \quad [y_{435},y_{3}]_c=0.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{34} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types super $CD$ and $A_1$.

 For $\xymatrix@R-8pt{  &  &  \overset{\zeta}{\circ} \ar  @{-}[d]^{\overline{\zeta}} & \\
	\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\
		}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}
	& \overset{-1}{\underset{\ }{\circ}}}$
the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
$y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;$
\begin{align*}
y_{1}^2&=\mu_1; & [y_{(13)},&y_2]_c=0; & y_{2}^2&=\mu_2; & y_{335}&=\lambda_{335}; \\
y_{332}&=0; & y_{4}^2&=\mu_4; & y_{334}&=0; &  y_{553}&=\lambda_{553}.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{34} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types super $A_4$ and $A_1$.

 For $\xymatrix@R-8pt{  &  &  \overset{\zeta}{\circ} \ar  @{-}[d]^{\overline{\zeta}} & \\
	\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\
		}{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}
	& \overset{-1}{\underset{\ }{\circ}}}$
the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
$y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;$
\begin{align*}
& & y_{2}^2&=\mu_2; & [[y_{43},&y_{435}]_c,y_3]_c=0; &
y_{3}^2&=\mu_3; & [y_{(24)},&y_3]_c=\nu_3;\\
y_{112}&=0; & y_{4}^2&=\mu_4; & [y_{(13)},&y_2]_c=0; & y_{553}&=0; & [y_{235},&y_3]_c=0.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{35} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types super $A_4$ and $A_1$.

 For $\xymatrix@R-8pt{  &  &  \overset{\zeta}{\circ} \ar  @{-}[d]^{\overline{\zeta}} & \\
	\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
		}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}
	& \overset{-1}{\underset{\ }{\circ}}}$
the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
$y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;$
\begin{align*}
y_{1}^2&=\mu_1; & y_{221}&=0; & y_{553}&=\lambda_{553}; & y_{332}&=\lambda_{332}; \\
y_{4}^2&=\mu_4; & y_{223}&=\lambda_{223};  & y_{334}&=0; & y_{335}&=\lambda_{335}.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{34} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types super $A_4$ and $A_1$.

 For $\xymatrix@R-8pt{  &   \overset{\zeta}{\circ} \ar  @{-}[d]^{\overline{\zeta}} & & \\
	\overset{\overline{\zeta}}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\
		}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}
	& \overset{\zeta}{\underset{\ }{\circ}}}$
the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
$y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;$
\begin{align*}
y_{552}&=0; & [y_{(13)},&y_2]_c=\nu_2; & y_{443}&=\lambda_{443}; & [[[y_{4325},&y_{2}]_c,y_3]_c,y_2]_c=0;
\\
y_{112}&=0; & [y_{125},&y_2]_c=\nu_2'; & y_{334}&=\lambda_{334}; & y_{332}&=0;
\quad  y_{2}^2=\mu_2.
\end{align*}
If $\nu_2'=0$, then $\widetilde{\mathcal E}_{\bq}/ \langle y_{34}\rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $A_4$, $A_1$. If $\nu_2'\neq 0$ but all the other $\lambda$'s are zero, then $\widetilde{\mathcal E}_{\bq}/ \langle y_{23}\rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types super $A_3$, $A_2$. The general case follows by Lemma \ref{lem:nonzero-k<l}.

\subsubsection{\ Type $\mathfrak{g}(8,3)$, $\zeta \in \mathrm{G}'_3$}\label{subsec:type-g(8,3)}
The Weyl groupoid has 21 objects:

 For $\xymatrix@C-4pt{\overset{-1}{\underset{\ }{\circ}}\ar  @{-}[r]^{\zeta}  &
	\overset{\overline{\zeta}}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{\overline{\zeta}}{\underset{\
		}{\circ}}\ar  @{-}[r]^{\overline{\zeta}}
	& \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
		}{\circ}}}$, the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
\begin{align*}
y_{221}&=0; & y_{223}&=\lambda_{223}; & y_{332}&=\lambda_{332}; & &y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
& & y_{443}&=0; & y_{445}&=\lambda_{445}; & &[y_{3345},y_{34}]_c=\lambda_{345};\\
& & y_{554}&=\lambda_{554}; & y_{1}^2&=\mu_1; & &[[y_{(24)},y_{3}]_c,y_3]_c=\nu_3;.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{12} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $F_4$ and $A_1$.

 For $\xymatrix@C-4pt{\overset{-1}{\underset{\ }{\circ}}\ar  @{-}[r]^{\overline{\zeta}}  &
	\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
		}{\circ}}
	\ar  @{-}[r]^{\overline{\zeta}}  & \overset{\overline{\zeta}}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  &
	\overset{-1}{\underset{\ }{\circ}}}$, the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
\begin{align*}
y_{221}&=0; & y_{223}&=\lambda_{223}; & y_{332}&=\lambda_{332}; & &y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
& & y_{334}&=0; & y_{445}&=0; & &[y_{4432},y_{43}]_c=\lambda_{432};\\
& & y_{1}^2&=\mu_1; & y_{5}^2&=\mu_5; & &[[y_{(35)},y_{4}]_c,y_4]_c=0;.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{45} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types super $B_4$ and $A_1$.

 For $\xymatrix@C-4pt{\overset{-1}{\underset{\ }{\circ}}\ar  @{-}[r]^{\zeta}  &
	\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
		}{\circ}}
	\ar  @{-}[r]^{\overline{\zeta}}  & \overset{\overline{\zeta}}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  &
	\overset{-1}{\underset{\ }{\circ}}}$, the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
\begin{align*}
[y_{(13)},&y_2]_c=0; & y_{332}&=0; & y_{334}&=0; & &y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
& & y_{445}&=0; & y_{1}^2&=\mu_1; & &[y_{4432},y_{43}]_c=0;\\
& & y_{2}^2&=\mu_2; & y_{5}^2&=\mu_5; & &[[y_{(35)},y_{4}]_c,y_4]_c=0.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{45} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types super $B_4$ and $A_1$.

 For $\xymatrix@C-4pt{\overset{-1}{\underset{\ }{\circ}}\ar  @{-}[r]^{\overline{\zeta}}  &
	\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
		}{\circ}}
	\ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  &
	\overset{-1}{\underset{\ }{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
\begin{align*}
& & y_{221}&=0; & y_{223}&=\lambda_{223}; & y_{334}&=0; & &y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{4}^2&=\mu_4; & y_{5}^2&=\mu_5; & y_{332}&=\lambda_{332}; & y_{1}^2&=\mu_1; & &[[y_{54},y_{543}]_c,y_4]_c=0.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{34} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types super $A_3$ and $A_2$.

 For $\xymatrix@C-4pt{\overset{\zeta}{\underset{\ }{\circ}}\ar  @{-}[r]^{\overline{\zeta}}  &
	\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\
		}{\circ}}
	\ar  @{-}[r]^{\overline{\zeta}}  & \overset{\overline{\zeta}}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  &
	\overset{-1}{\underset{\ }{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
\begin{align*}
y_{112}&=0; & [y_{443},&y_{43}]_c=\lambda_{43}; & y_{445}&=0; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
& & [y_{(13)},&y_2]_c=0; & y_{2}^2&=\mu_2; & &[y_{4432},y_{43}]_c=0;\\
y_{5}^2&=\mu_5; & [y_{(24)},&y_3]_c=0; & y_{3}^2&=\mu_3; & &[[y_{(35)},y_4]_c,y_4]_c=\lambda_{(35)}.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{23} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types super $A_2$ and $\mathfrak{g}(2,3)$.

 For $\xymatrix@C-4pt{\overset{-1}{\underset{\ }{\circ}}\ar  @{-}[r]^{\zeta}  &
	\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
		}{\circ}}
	\ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  &
	\overset{-1}{\underset{\ }{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
\begin{align*}
y_{5}^2&=\mu_5; & y_{332}&=0; & y_{334}&=0; & y_{1}^2&=\mu_1; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
& & [y_{(13)},&y_2]_c=0; & y_{2}^2&=\mu_2; & y_{4}^2&=\mu_4; & &[[y_{54},y_{543}]_c,y_4]_c=0.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{34} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types super $A_2$ and $A_3$.

 For $\xymatrix@C-4pt{\overset{\zeta}{\underset{\ }{\circ}}\ar  @{-}[r]^{\overline{\zeta}}  &
	\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\
		}{\circ}}
	\ar  @{-}[r]^{\zeta}  & \overset{-\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  &
	\overset{-1}{\underset{\ }{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
\begin{align*}
y_{3}^2&=\mu_3; & y_{221}&=\lambda_{221}; & y_{4443}&=\lambda_{4443}; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{5}^2&=\mu_5; & y_{112}&=\lambda_{112}; & y_{4445}&=\lambda_{4445}; & &[y_{(24)},y_{3}]_c=0;
\end{align*}
\vspace*{-0.7cm}
\begin{align*}
y_{223}&=0; & [y_3,y_{445}]_c+q_{45}[y_{(35)},y_4]_c+(\zeta-\overline{\zeta})q_{34}y_4y_{(35)}=0.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{23} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types super $A_2$ and $\mathfrak{g}(2,3)$.

 For $\xymatrix@C-4pt{\overset{\zeta}{\underset{\ }{\circ}}\ar  @{-}[r]^{\overline{\zeta}}  &
	\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\
		}{\circ}}\ar  @{-}[r]^{\zeta}
	& \overset{\overline{\zeta}}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\
		}{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
\begin{align*}
y_{3}^2&=\mu_3; & [y_{(24)},&y_3]_c=\nu_3; & y_{112}&=\lambda_{112}; & y_{223}&=0; \\
y_{5}^2&=\mu_5; & [y_{445},&y_{45}]_c=\lambda_{45}; & y_{221}&=\lambda_{221}; & y_{445}&=0;
\end{align*}
\vspace*{-0.7cm}
\begin{align*}
 y_{ij}&=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1; & [[y_{(35)},y_{4}]_c,y_4]_c&=\lambda_{(35)}.
\end{align*}
If $\nu_3=0$, then $\widetilde{\mathcal E}_{\bq}/ \langle y_{23} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types super $A_2$ and $\mathfrak{g}(2,3)$. If $\nu_3\neq 0$, then $\lambda_{(35)}= \lambda_{45}=0$, so $\widetilde{\mathcal E}_{\bq}/ \langle y_{45} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types super $A_4$ and $A_1$.

 For $\xymatrix@C-4pt{\overset{\zeta}{\underset{\ }{\circ}}\ar  @{-}[r]^{\overline{\zeta}}  &
	\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\
		}{\circ}}\ar  @{-}[r]^{\overline{\zeta}}
	& \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\
		}{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
\begin{align*}
y_{4}^2&=\mu_4; & y_{5}^2&=\mu_5; & [y_{(24)},&y_3]_c=\nu_3; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
&& y_{2}^2&=\mu_2; &  [y_{(13)},&y_2]_c=0; &   &[[y_{34},y_{(35)}]_c,y_4]_c=\lambda_{345};\\
&& y_{3}^2&=\mu_3; & y_{112}&=0; & &[[y_{54},y_{543}]_c,y_4]_c=\lambda_{543}.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{12} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $A_1$ and $\mathfrak{g}(4,3)$.

 For $\xymatrix@C-4pt{\overset{\overline{\zeta}}{\underset{\ }{\circ}}\ar  @{-}[r]^{\zeta}  &
	\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\
		}{\circ}}\ar  @{-}[r]^{\overline{\zeta}}
	& \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
		}{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
\begin{align*}
[y_{(13)},&y_2]_c=0; & y_{2}^2&=\mu_2; & y_{112}&=0; & y_{445}&=\lambda_{445}; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
& & y_{3}^2&=\mu_3; &  y_{443}&=0; & y_{554}&=\lambda_{554}; & &[[y_{23},y_{(24)}]_c,y_3]_c=0.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{23} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types super $A_2$ and $A_3$.

 For $\xymatrix@C-4pt{\overset{-1}{\underset{\ }{\circ}}\ar  @{-}[r]^{\overline{\zeta}}  &
	\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{\overline{\zeta}}{\underset{\
		}{\circ}}\ar  @{-}[r]^{\overline{\zeta}}
	& \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
		}{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
\begin{align*}
y_{1}^2&=\mu_1; & [y_{(13)},&y_2]_c=0; & y_{332}&=0; & y_{445}&=\lambda_{445};\\
y_{2}^2&=\mu_2; & [y_{3345},&y_{34}]_c=\lambda_{(35)}'; & y_{443}&=0; & y_{554}&=\lambda_{554};
\end{align*}
\vspace*{-0.7cm}
\begin{align*}
 y_{ij}&=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1; & [[y_{(35)},y_{4}]_c,y_4]_c&=\lambda_{(35)}.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{23} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types super $A_2$ and $C_3$.

 For $\xymatrix@R-8pt{  &   & \overset{-1}{\circ} \ar  @{-}[d]^{\zeta}\ar@{-}[dr]^{\zeta}  & \\
	\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
		}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}
	& \overset{\overline{\zeta}}{\underset{\ }{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
\begin{align*}
y_1^2&=\mu_1; & [y_{(24)},&y_3]_c=\nu_3; & y_{221}&=0; & y_{223}&=0; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_3^2&=\mu_3; & y_5^2&=\mu_5; & y_{443}&=0; & y_{445}&=0; & &[y_{235},y_3]_c=0;
\end{align*}
\vspace*{-0.7cm}
\begin{align*}
y_{(35)}-q_{45}\overline{\zeta}[y_{35},y_4]_c-q_{34}(1-\zeta)y_4y_{35}=0.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{34},y_{45} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types super $A_4$ and $A_1$.

 For $\xymatrix@R-8pt{  &   & \overset{-1}{\circ} \ar  @{-}[d]^{\zeta} \ar@{-}[dr]^{\zeta} &
	\\
	\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\
		}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}
	& \overset{\overline{\zeta}}{\underset{\ }{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
\begin{align*}
[y_{(13)},&y_2]_c=\nu_2; & y_{443}&=0; & y_1^2&=\mu_1; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
[y_{(24)},&y_3]_c=0; & y_{445}&=0; & y_2^2&=\mu_2; & &[y_{235},y_3]_c=\nu_3';
\end{align*}
\vspace*{-0.7cm}
\begin{align*}
y_3^2&=\mu_3; & y_5^2&=\mu_5; & y_{(35)}-q_{45}\overline{\zeta}[y_{35},y_4]_c-q_{34}(1-\zeta)y_4y_{35}=0.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{34},y_{45} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types super $A_4$ and $A_1$.

 For $\xymatrix@R-8pt{  &    \overset{-1}{\circ} \ar  @{-}[d]^{\zeta} \ar  @{-}[dr]^{\zeta}
	&  & \\
	\overset{\overline{\zeta}}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{\overline{\zeta}}{\underset{\
		}{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}
	& \overset{\zeta}{\underset{\ }{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
\begin{align*}
[y_{(24)},&y_3]_c=\nu_3; & y_{112}&=\lambda_{112}; & y_{223}&=0; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{443}&=0; & y_{221}&=\lambda_{221}; & y_{225}&=0; & &[y_{435},y_3]_c=0;
\end{align*}
\vspace*{-0.7cm}
\begin{align*}
y_3^2&=\mu_3; & y_5^2&=\mu_5; & y_{235}-q_{35}\overline{\zeta}[y_{25},y_3]_c-q_{23}(1-\zeta)y_3y_{25}=0.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{25},y_{35} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types super $A_4$ and $A_1$.

 For $\xymatrix@R-8pt{  &   & \overset{-1}{\circ} \ar  @{-}[d]^{\zeta} \ar@{-}[dr]^{\zeta} &
	\\
	\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\
		}{\circ}} \ar  @{-}[r]^{\zeta}  &
	\overset{\overline{\zeta}}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{\overline{\zeta}}{\underset{\
		}{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
\begin{align*}
[y_{(13)},&y_2]_c=\nu_2; & y_{112}&=0; & y_{332}&=0; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{445}&=0; & y_{334}&=\lambda_{334}; & y_{335}&=0; & &y_{443}=\lambda_{443};
\end{align*}
\vspace*{-0.7cm}
\begin{align*}
y_2^2&=\mu_2; & y_5^2&=\mu_5; &
y_{(35)}-q_{45}\overline{\zeta}[y_{35},y_4]_c-q_{34}(1-\zeta)y_4y_{35}=0.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{35},y_{45} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types super $A_4$ and $A_1$.

 For $\xymatrix@R-8pt{  &   & \overset{\zeta}{\circ} \ar  @{-}[d]^{\overline{\zeta}}  & \\
	\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{\overline{\zeta}}{\underset{\
		}{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}
	& \overset{-1}{\underset{\ }{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
\begin{align*}
y_{3}^2&=\mu_3; & [y_{(24)},&y_3]_c=0; & y_{221}&=0; & y_{223}&=0; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{4}^2&=\mu_4; & [y_{235},&y_3]_c=\nu_3'; &  y_{553}&=0; & y_1^2&=\mu_1; & &[[y_{43},y_{435}]_c,y_3]_c=0.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{35} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types super $A_4$ and $A_1$.

 For $\xymatrix@R-8pt{  &   & \overset{\zeta}{\circ} \ar  @{-}[d]^{\overline{\zeta}}  & \\
	\overset{\overline{\zeta}}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\
		}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}
	& \overset{-1}{\underset{\ }{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
\begin{align*}
y_{112}&=0; & y_{2}^2&=\mu_2; & y_{332}&=0; & y_{335}&=\lambda_{335}; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
& & y_{4}^2&=\mu_4; & y_{334}&=0; & y_{553}&=\lambda_{553}; & &[y_{(13)},y_2]_c=\nu_2.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{34} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types super $A_4$ and $A_1$.

 For $\xymatrix@R-8pt{  &   & \overset{\zeta}{\circ} \ar  @{-}[d]^{\overline{\zeta}}  & \\
	\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\
		}{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{\overline{\zeta}}{\underset{\ }{\circ}} \ar
	@{-}[r]^{\zeta}  & \overset{-1}{\underset{\ }{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
\begin{align*}
y_1^2&=\mu_1; & [[y_{235},&y_3]_c,y_3]_c=0; & [y_{(13)},&y_2]_c=0; &  y_{ij}&=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{2}^2&=\mu_2; & [[y_{435},&y_3]_c,y_3]_c=0; &  y_4^2&=\mu_4; & y_{332}&=y_{334}=y_{553}=0.
\end{align*}
This algebra is nonzero by \cite[Lemma 5.16]{AAGMV}.

 For $\xymatrix@R-8pt{  &   & \overset{\zeta}{\circ} \ar  @{-}[d]^{\overline{\zeta}}  & \\
	\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{\overline{\zeta}}{\underset{\
		}{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{\overline{\zeta}}{\underset{\ }{\circ}} \ar
	@{-}[r]^{\zeta}  & \overset{-1}{\underset{\ }{\circ}}}$
the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
\begin{align*}
y_{1}^2&=\mu_1; & y_{221}&=y_{334}=y_{553}=0; & y_{223}&=\lambda_{223}; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{4}^2&=\mu_4; & [[y_{435},&y_3]_c,y_3]_c=0; & y_{332}&=\lambda_{332}; &  &[[y_{235},y_3]_c,y_3]_c=\lambda_{235}.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{34} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types super $CD$ and $A_1$.

 For $\xymatrix@R-8pt{  &   & \overset{\zeta}{\circ} \ar  @{-}[d]^{\overline{\zeta}}  & \\
	\overset{\overline{\zeta}}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\
		}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}
	& \overset{-1}{\underset{\ }{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
\begin{align*}
y_{112}&=0; & [y_{(24)},&y_3]_c=\nu_3; & y_{4}^2&=\mu_4; &  & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
&&     [y_{(13)},&y_2]_c=0; &  y_{3}^2&=\mu_3; &   &[[y_{23},y_{235}]_c,y_3]_c=0;\\
&& y_{553}&=0; & y_{2}^2&=\mu_2; & &[y_{435},y_3]_c=0.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{35} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types super $A_4$ and $A_1$.

 For $\xymatrix@R-8pt{  &   & \overset{\zeta}{\circ} \ar  @{-}[d]^{\overline{\zeta}}  & \\
	\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\
		}{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}
	& \overset{-1}{\underset{\ }{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
\begin{align*}
y_{1}^2&=\mu_1; & [y_{(24)},&y_3]_c=\nu_3; & y_{553}&=0; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
&&     [y_{(13)},&y_2]_c=\nu_2; &  y_{4}^2&=\mu_4; &   &[[y_{43},y_{435}]_c,y_3]_c=0;\\
&& y_{3}^2&=\mu_3; & y_{2}^2&=\mu_2; & &[y_{235},y_{3}]_c=0.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{35} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types super $A_4$ and $A_1$.

\subsubsection{Type $\mathfrak{g}(4,6)$, $\zeta \in \mathrm{G}'_3$}\label{subsec:type-g(4,6)}
The Weyl groupoid has seven objects.

 For $\xymatrix@C-4pt{\overset{\zeta}{\underset{\ }{\circ}}\ar  @{-}[r]^{\overline{\zeta}}  &
\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
}{\circ}}
\ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  &
\overset{\zeta}{\underset{\ }{\circ}}  \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
}{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_6$, with defining relations
\begin{align*}
y_{112}&=\lambda_{112}; & y_{221}&=\lambda_{221}; & y_{223}&=\lambda_{223}; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{665}&=\lambda_{665}; & y_{332}&=\lambda_{332}; & y_{334}&=0; & &[[[y_{(25)},y_{4}]_c,y_3]_c,y_4]_c=0;\\
y_{4}^2&=\mu_4; & y_{554}&=0; & y_{556}&=\lambda_{556}; & &[[[y_{6543},y_{4}]_c,y_5]_c,y_4]_c=0.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{34} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $A_3$.

 For $\xymatrix@C-5pt@R-8pt{  &   & & \overset{\zeta}{\circ} \ar  @{-}[d]^{\overline{\zeta}}  & \\
\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}
 & \overset{-1}{\underset{\ }{\circ}}
\ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\ }{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_6$, with defining relations
\begin{align*}
y_{4}^2&=\mu_4; & y_{221}&=\lambda_{221}; & y_{223}&=\lambda_{223}; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{5}^2&=\mu_5; & y_{334}&=0; & y_{664}&=0; & &[y_{546},y_{4}]_c=0;\\
[y_{(35)},y_{4}]_c&=0; & y_{112}&=\lambda_{112}; & y_{332}&=\lambda_{332}; & &[[[y_{2346},y_4]_c,y_3]_c,y_4]_c=0.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{45} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $CD$ and $A_1$.

 For $\xymatrix@C-5pt@R-8pt{  &   & \overset{\zeta}{\circ} \ar  @{-}[d]^{\overline{\zeta}}  & &\\
\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\
}{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}
& \overset{\zeta}{\underset{\ }{\circ}}
\ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_6$, with defining relations
\begin{align*}
[y_{(13)},&y_{2}]_c=0; & y_{445}&=\lambda_{445}; & y_{443}&=0; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{112}&=0; & y_{554}&=\lambda_{554}; & y_{663}&=0; & &[y_{236},y_{3}]_c=0;\\
[y_{(24)},&y_{3}]_c=0; & y_{2}^2&=\mu_2; & y_{3}^2&=\mu_3; & &[[[y_{5436},y_3]_c,y_4]_c,y_3]_c=0.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{34} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $A_4$ and $A_2$.

 For $\xymatrix@C-5pt@R-8pt{  &   & & \overset{\zeta}{\circ} \ar  @{-}[d]^{\overline{\zeta}}  & \\
\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}
 & \overset{\zeta}{\underset{\ }{\circ}}
\ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\ }{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_6$, with defining relations
\begin{align*}
y_{112}&=\lambda_{112}; & y_{221}&=\lambda_{221}; & y_{223}&=\lambda_{223}; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{332}&=\lambda_{332}; & y_{334}&=\lambda_{334}; & y_{443}&=\lambda_{443}; & & y_{445}=0;\\
y_{446}&=\lambda_{446}; & y_{664}&=\lambda_{664}; & y_{5}^2&=\mu_5.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{45} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $A_5$ and $A_1$.

 For $\xymatrix@C-5pt@R-8pt{  &   & \overset{\zeta}{\circ} \ar  @{-}[d]^{\overline{\zeta}}  & &\\
\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\
}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}
 & \overset{\zeta}{\underset{\ }{\circ}}
\ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_6$, with defining relations
\begin{align*}
y_{332}&=0; & y_{334}&=\lambda_{334}; & y_{336}&=\lambda_{336}; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{663}&=\lambda_{663}; & y_{443}&=\lambda_{443}; & y_{445}&=\lambda_{443}; & &[y_{(13)},y_2]_c=0;\\
y_{554}&=\lambda_{554}; & y_{1}^2&=\mu_1; & y_{2}^2&=\mu_2.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{23} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $A_2$ and $A_4$.

 For $\xymatrix@C-5pt@R-8pt{  &    & \overset{\zeta}{\circ} \ar  @{-}[d]^{\overline{\zeta}} &  & \\
\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}
 & \overset{\zeta}{\underset{\ }{\circ}}
\ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\ }{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_6$, with defining relations
\begin{align*}
y_{221}&=0; & y_{223}&=\lambda_{223}; & y_{332}&=\lambda_{332}; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{334}&=\lambda_{334}; & y_{443}&=\lambda_{443}; & y_{445}&=0; & & y_{336}=\lambda_{336};\\
y_{663}&=\lambda_{663}; & y_{1}^2&=\mu_1; & y_{5}^2&=\mu_5.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{45} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $CD$ and $A_1$.

 For $\xymatrix@C-5pt@R-8pt{  &   & \overset{-1}{\circ} \ar  @{-}[d]^{\zeta} \ar
@{-}[dr]^{\zeta}  & &\\
\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}
& \overset{-1}{\underset{\ }{\circ}}
\ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}}}$
the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_6$, with defining relations
\begin{align*}
y_{112}&=\lambda_{112}; & y_{6}^2&=\mu_6; & y_{223}&=0; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{554}&=0; & y_{4}^2&=\mu_4; & [y_{(24)},&y_3]_c=0; &  &[y_{236},y_3]_c=0;\\
y_{221}&=\lambda_{221}; & y_{3}^2&=\mu_3; & [y_{(35)},&y_4]_c=0; & &[y_{546},y_4]_c=0;
\end{align*}
\vspace*{-0.7cm}
\begin{align*}
y_{346}-q_{46}\overline{\zeta} [y_{36},y_4]_c-q_{34}(1-\zeta)y_4y_{36}=0.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{36}, y_{46} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $A_5$ and $A_1$.

\subsubsection{Type $\mathfrak{g}(6,6)$, $\zeta \in \mathrm{G}'_3$}\label{subsec:type-g(6,6)}
The Weyl groupoid has 21 objects.

 For $\xymatrix@C-6pt{\overset{\zeta}{\underset{\ }{\circ}}\ar  @{-}[r]^{\overline{\zeta}}  &
\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
}{\circ}}
\ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  &
\overset{\overline{\zeta}}{\underset{\ }{\circ}}  \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\
}{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_6$, with defining relations
\begin{align*}
y_{112}&=\lambda_{112}; & y_{221}&=\lambda_{221}; & y_{223}&=\lambda_{223}; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{556}&=0 & y_{332}&=\lambda_{332}; & y_{334}&=\lambda_{334}; &  &[[y_{(46)},y_5]_c,y_5]_c=0;\\
y_{6}^2&=\mu_6; & y_{443}&=\lambda_{443}; & y_{445}&=\lambda_{445}; & &[y_{5543},y_{54}]_c=\lambda_{543}.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{56} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $B_5$ and $A_1$.

 For $\xymatrix@C-6pt{\overset{\zeta}{\underset{\ }{\circ}}\ar  @{-}[r]^{\overline{\zeta}}  &
\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
}{\circ}}
\ar  @{-}[r]^{\overline{\zeta}}    & \overset{\zeta}{\underset{\ }{\circ}}  \ar  @{-}[r]^{\overline{\zeta}} &
\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\
}{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_6$, with defining relations
\begin{align*}
y_{5}^2&=\mu_5; & y_{112}&=\lambda_{112}; & y_{221}&=\lambda_{221}; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{6}^2&=\mu_6; & y_{223}&=\lambda_{223}; & y_{332}&=\lambda_{332}; &  &[[y_{65},y_{654}]_c,y_5]_c=0;\\
& & y_{334}&=\lambda_{334}; & y_{443}&=\lambda_{443}; & &y_{445}=0.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{56} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $A_5$ and $A_1$.

 For $\xymatrix@C-7pt{\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  &
\overset{-1}{\underset{\ }{\circ}}\ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\
}{\circ}}\ar  @{-}[r]^{\overline{\zeta}}  &
\overset{-1}{\underset{\ }{\circ}}\ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
}{\circ}}
\ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_6$, with defining relations
\begin{align*}
y_{112}&=0; & y_{2}^2&=\mu_2; & y_{554}&=0; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{556}&=\lambda_{556}; & y_{3}^2&=\mu_3; & y_{665}&=\lambda_{665}; & &[[y_{34},y_{(35)}]_c,y_4]_c=0;\\
& & y_{4}^2&=\mu_4; & [y_{(13)},&y_2]_c=0; & &[y_{(24)},y_3]_c=\nu_3.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{45} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $A_4$ and $A_2$.

 For $\xymatrix@C-7pt{\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  &
\overset{-1}{\underset{\ }{\circ}}
\ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  &
\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
}{\circ}}\ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_6$, with defining relations
\begin{align*}
y_1^2&=\mu_1; & y_{334}&=0; & y_{556}&=\lambda_{556}; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_2^2&=\mu_2; & y_{554}&=0; & y_{665}&=\lambda_{665}; &  &[[[y_{(25)},y_4]_c,y_3]_c,y_4]_c=\lambda_{_{(25)}};\\
y_4^2&=\mu_4; & y_{332}&=0; & [y_{(13)},&y_2]_c=0; & &[[[y_{6543},y_4]_c,y_5]_c,y_4]_c=0.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{12} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $\mathfrak{el}(5,3)$ and $A_1$.

 For $\xymatrix@C-6pt{\overset{-1}{\underset{\ }{\circ}}\ar  @{-}[r]^{\overline{\zeta}}  &
\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
}{\circ}}
\ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  &
\overset{\zeta}{\underset{\ }{\circ}}\ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
}{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_6$, with defining relations
\begin{align*}
y_{221}&=0; & y_{223}&=\lambda_{223}; & y_{332}&=\lambda_{332}; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{1}^2&=\mu_1; & y_{334}&=0; & y_{554}&=0; & &[[[y_{(25)},y_4]_c,y_3]_c,y_4]_c=0;\\
y_{4}^2&=\mu_4; & y_{556}&=\lambda_{556}; & y_{665}&=\lambda_{665}; & &[[[y_{6543},y_4]_c,y_5]_c,y_4]_c=0.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{45} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $A_4$ and $A_2$.

 For $\xymatrix@C-7pt{\overset{\zeta}{\underset{\ }{\circ}}\ar  @{-}[r]^{\overline{\zeta}}  &
\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\
}{\circ}}
\ar  @{-}[r]^{\zeta}  & \overset{\overline{\zeta}}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  &
\overset{\zeta}{\underset{\ }{\circ}}\ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
}{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_6$, with defining relations
\begin{align*}
y_{3}^2&=\mu_3; & y_{112}&=\lambda_{112}; & y_{221}&=\lambda_{221}; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{665}&=\lambda_{665}; & y_{223}&=0; & y_{443}&=0; &  &[[y_{(35)},y_4]_c,y_4]_c=0;\\
[y_{(24)},y_3]_c&=\nu_2; & y_{554}&=0; & y_{556}&=\lambda_{665}; & &[y_{4456},y_{45}]_c=\lambda_{456}.
\end{align*}
If $\nu_2=0$, then $\widetilde{\mathcal E}_{\bq}/ \langle y_{34} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $A_3$ and $B_3$. If $\nu_2\neq 0$, then $\lambda_{456}=0$, so $\widetilde{\mathcal E}_{\bq}/ \langle y_{45} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $A_4$ and $A_2$.

 For $\xymatrix@R-8pt{  &  & & \overset{-1}{\circ} \ar  @{-}[d]^{\zeta}\ar@{-}[dr]^{\zeta}  & \\
\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}} & \overset{\zeta}{\underset{\
}{\circ}} \ar  @{-}[r]^{\overline{\zeta}} & \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}
& \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{\overline{\zeta}}{\underset{\
}{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_6$, with defining relations
\begin{align*}
y_{112}&=\lambda_{112}; & y_{221}&=\lambda_{221}; & y_{223}&=\lambda_{223}; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{4}^2&=\mu_4; & y_{332}&=\lambda_{332}; & y_{334}&=0; & &[y_{346},y_4]_c=0;\\
y_{5}^2&=\mu_5; & y_{554}&=0; & y_{556}&=0; & &[y_{(35)},y_4]_c=\nu_4;
\end{align*}
\vspace*{-0.7cm}
\begin{align*}
y_{(46)}-q_{56}\overline{\zeta} [y_{46},y_5]_c-q_{45}(1-\zeta)y_5y_{46}=0.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{46},y_{56} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $A_5$ and $A_1$.

 For $\xymatrix@C-3pt@R-8pt{  &   & \overset{-1}{\circ} \ar  @{-}[d]^{\zeta} \ar
@{-}[dr]^{\zeta}  & \\
\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}
& \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}
& \overset{-1}{\underset{\ }{\circ}}\ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
}{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_6$, with defining relations
\begin{align*}
y_{221}&=0; & y_{223}&=0; & y_{554}&=0; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{4}^2&=\mu_4; & y_{1}^2&=\mu_1; & [y_{(24)},&y_3]_c=0; & &[y_{236},y_3]_c=0;\\
y_{6}^2&=\mu_6; & y_{3}^2&=\mu_3; & [y_{(35)},&y_4]_c=0; & &[y_{546},y_4]_c=0;
\end{align*}
\vspace*{-0.7cm}
\begin{align*}
y_{346}-q_{46}\overline{\zeta} [y_{36},y_4]_c-q_{34}(1-\zeta)y_4y_{36}=0.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{36},y_{46} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $A_5$ and $A_1$.

 For $\xymatrix@C-3pt@R-8pt{  &   & \overset{-1}{\circ} \ar  @{-}[d]^{\zeta} \ar
@{-}[dr]^{\zeta}  & \\
\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\
}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}
& \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}
& \overset{-1}{\underset{\ }{\circ}}\ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
}{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_6$, with defining relations
\begin{align*}
y_{554}&=0; & y_{1}^2&=\mu_1; & [y_{(13)},&y_2]_c=\nu_2; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{4}^2&=\mu_4; & y_{2}^2&=\mu_2; & [y_{(24)},&y_3]_c=\nu_3; &  &[y_{236},y_3]_c=\nu_3';\\
y_{6}^2&=\mu_6; & y_{3}^2&=\mu_3; & [y_{(35)},&y_4]_c=0; & &[y_{546},y_4]_c=0;
\end{align*}
\vspace*{-0.7cm}
\begin{align*}
y_{346}-q_{46}\overline{\zeta} [y_{36},y_4]_c-q_{34}(1-\zeta)y_4y_{36}=0.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{45} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $CD$ and $A_1$.

 For $\xymatrix@C-3pt@R-8pt{  &   & \overset{-1}{\circ} \ar  @{-}[d]^{\zeta} \ar
@{-}[dr]^{\zeta}  & \\
\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\
}{\circ}} \ar  @{-}[r]^{\zeta}
& \overset{\overline{\zeta}}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}
& \overset{-1}{\underset{\ }{\circ}}\ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
}{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_6$, with defining relations
\begin{align*}
y_{112}&=0; & y_{332}&=0; & y_{334}&=0; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{4}^2&=\mu_4; & y_{336}&=0; & [y_{(13)},&y_2]_c=\nu_2; & &[y_{(35)},y_4]_c=\nu_4;\\
y_{6}^2&=\mu_6; & y_{554}&=0; & y_2^2&=\mu_2; & &[y_{546},y_4]_c=0;
\end{align*}
\vspace*{-0.7cm}
\begin{align*}
y_{346}-q_{46}\overline{\zeta} [y_{36},y_4]_c-q_{34}(1-\zeta)y_4y_{36}=0.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{36},y_{46} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $A_5$ and $A_1$.

 For $\xymatrix@C-3pt@R-8pt{  & &  & \overset{\zeta}{\circ} \ar  @{-}[d]^{\overline{\zeta}}  & \\
\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}} & \overset{\zeta}{\underset{\
}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}
& \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\
}{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_6$, with defining relations
\begin{align*}
y_{112}&=\lambda_{112}; & y_{221}&=\lambda_{221}; & y_{223}&=0; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{4}^2&=\mu_4; & y_{664}&=0; & [y_{(24)},&y_3]_c=0; & &[[y_{54},y_{546}]_c,y_4]_c=0;\\
y_{5}^2&=\mu_5; & y_{3}^2&=\mu_3; & [y_{(35)},&y_4]_c=\nu_4; & &[y_{346},y_4]_c=0.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{46} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $A_5$ and $A_1$.

 For $\xymatrix@R-8pt{  & &  & \overset{\zeta}{\circ} \ar  @{-}[d]^{\overline{\zeta}}  & \\
\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}} & \overset{-1}{\underset{\
}{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}
& \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\
}{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_6$, with defining relations
\begin{align*}
y_{3}^2&=\mu_3; & y_{112}&=0; & y_{443}&=0; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{5}^2&=\mu_5; & y_{445}&=0; & y_{446}&=\lambda_{446}; & &[y_{(13)},y_2]_c=0;\\
& & y_{664}&=\lambda_{664}; & y_2^2&=\mu_2; & &[y_{(24)},y_3]_c=0.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{34} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of type $A_3$.

 For $\xymatrix@R-8pt{  & &  & \overset{\zeta}{\circ} \ar  @{-}[d]^{\overline{\zeta}}  & \\
\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}} & \overset{\zeta}{\underset{\
}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}
& \overset{\overline{\zeta}}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\
}{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_6$, with defining relations
\begin{align*}
y_{112}&=\lambda_{112}; & y_{221}&=\lambda_{221}; & y_{223}&=0; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{3}^2&=\mu_3; & y_{443}&=0; & [y_{(24)},&y_3]_c=\nu_3; & &[[y_{346},y_4]_c,y_4]_c=0;\\
y_{5}^2&=\mu_5; & y_{445}&=0; & y_{664}&=0; & &[[y_{546},y_4]_c,y_4]_c=0.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{46} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $A_5$ and $A_1$.

 For $\xymatrix@R-8pt{  & &  & \overset{\zeta}{\circ} \ar  @{-}[d]^{\overline{\zeta}}  & \\
\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta} & \overset{-1}{\underset{\
}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}
& \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\
}{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_6$, with defining relations
\begin{align*}
y_1^2&=\mu_1; & y_{332}&=0; & y_{334}&=\lambda_{334}; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_2^2&=\mu_2; & y_{443}&=\lambda_{443}; & y_{445}&=0; & &[y_{(13)},y_2]_c=0;\\
y_5^2&=\mu_5; & y_{446}&=\lambda_{446}; & y_{664}&=\lambda_{664}. & &
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{45} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $A_5$ and $A_1$.

 For $\xymatrix@C-4pt@R-8pt{  & &  & \overset{\zeta}{\circ} \ar  @{-}[d]^{\overline{\zeta}}  & \\
\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}} & \overset{-1}{\underset{\
}{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}
& \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\
}{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_6$, with defining relations
\begin{align*}
y_{112}&=0; & y_{664}&=0; & [y_{(13)},&y_2]_c=0; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{4}^2&=\mu_4; & y_{2}^2&=\mu_2; & [y_{(24)},&y_3]_c=\nu_3; & &[[y_{34},y_{346}]_c,y_4]_c=0;\\
y_{5}^2&=\mu_5; & y_{3}^2&=\mu_3; & [y_{(35)},&y_4]_c=\nu_4; & &[y_{546},y_4]_c=0.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{46} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $A_5$ and $A_1$.

 For $\xymatrix@R-8pt{  & &  & \overset{\zeta}{\circ} \ar  @{-}[d]^{\overline{\zeta}}  & \\
\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}} & \overset{\zeta}{\underset{\
}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}
& \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\
}{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_6$, with defining relations
\begin{align*}
y_{1}^2&=\mu_1; & y_{221}&=0; & y_{223}&=\lambda_{223}; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{5}^2&=\mu_5; & y_{332}&=\lambda_{332}; & y_{334}&=\lambda_{334}; &  &y_{443}=\lambda_{443}; \\
& & y_{445}&=0; & y_{446}&=\lambda_{446}; & &y_{664}=\lambda_{664}.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{45} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $A_5$ and $A_1$.

 For $\xymatrix@C-4pt@R-8pt{  & &  & \overset{\zeta}{\circ} \ar  @{-}[d]^{\overline{\zeta}}  & \\
\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta} & \overset{-1}{\underset{\
}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}
& \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\
}{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_6$, with defining relations
\begin{align*}
y_{1}^2&=\mu_1; & y_{332}&=0; & y_{334}&=0; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{2}^2&=\mu_2; & y_{664}&=0; & [y_{(13)},&y_2]_c=0; & &[y_{564},y_4]_c=0;\\
& & y_{4}^2&=\mu_4; & y_{5}^2&=\mu_5; & &[y_{(35)},y_4]_c=0.
\end{align*}
This algebra is nonzero by \cite[Lemma 5.16]{AAGMV}.

 For $\xymatrix@C-4pt@R-8pt{  &   & \overset{\zeta}{\circ} \ar  @{-}[d]^{\overline{\zeta}}  & \\
\overset{\overline{\zeta}}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\
}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}
& \overset{\zeta}{\underset{\ }{\circ}}\ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
}{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_6$, with defining relations
\begin{align*}
y_{2}^2&=\mu_2; & y_{112}&=0; & y_{332}&=0; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{554}&=\lambda_{554}; & y_{334}&=\lambda_{334}; & y_{336}&=\lambda_{336}; & &[y_{(13)},y_2]_c=\nu_2;\\
& & y_{663}&=\lambda_{663}; & y_{443}&=\lambda_{443}; & &y_{445}=\lambda_{445}.
\end{align*}
If $\nu_2=0$, then $\widetilde{\mathcal E}_{\bq}/ \langle y_{23} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $A_2$ and $A_4$. If $\nu_2\neq 0$, then $\lambda_{336}=\lambda_{663}=0$, so $\widetilde{\mathcal E}_{\bq}/ \langle y_{36} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $A_5$ and $A_1$.

 For $\xymatrix@C-3pt@R-8pt{  & &  & \overset{\zeta}{\circ} \ar  @{-}[d]^{\overline{\zeta}}  & \\
\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}} & \overset{\zeta}{\underset{\
}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}
& \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\
}{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_6$, with defining relations
\begin{align*}
y_{1}^2&=\mu_1; & y_{223}&=\lambda_{223}; & y_{334}&=0; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{4}^2&=\mu_4; & y_{332}&=\lambda_{332}; & [y_{(35)},&y_4]_c=0; & &[[[y_{2346},y_4]_c,y_3]_c,y_4]_c=0;\\
y_{5}^2&=\mu_5; & y_{664}&=0; & y_{221}&=0; & &[y_{546},y_4]_c=0.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{34} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of type $A_3$.

 For $\xymatrix@C-3pt@R-8pt{  &   & \overset{\zeta}{\circ} \ar  @{-}[d]^{\overline{\zeta}}  & \\
\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{\overline{\zeta}}{\underset{\
}{\circ}} \ar  @{-}[r]^{\zeta}
& \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}
& \overset{\zeta}{\underset{\ }{\circ}}\ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
}{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_6$, with defining relations
\begin{align*}
y_{221}&=0; & y_{223}&=0; & y_{443}&=0; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{1}^2&=\mu_1; & y_{445}&=\lambda_{445}; & [y_{(24)},&y_3]_c=0; & &[[[y_{5436},y_3]_c,y_4]_c,y_3]_c=0;\\
y_{3}^2&=\mu_3; & y_{554}&=\lambda_{554}; & y_{663}&=0; & &[y_{236},y_3]_c=\nu_3'.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{34} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $A_4$ and $A_2$.

 For $\xymatrix@C-3pt@R-8pt{  &   & \overset{\zeta}{\circ} \ar  @{-}[d]^{\overline{\zeta}}   & \\
\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\
}{\circ}} \ar  @{-}[r]^{\zeta}
& \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}
& \overset{\zeta}{\underset{\ }{\circ}}\ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
}{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_6$, with defining relations
\begin{align*}
y_{1}^2&=\mu_1; & y_{445}&=\lambda_{445}; & y_{554}&=\lambda_{554}; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{2}^2&=\mu_2; & y_{663}&=0; & [y_{(13)},&y_2]_c=\nu_2; & &[[[y_{5436},y_3]_c,y_4]_c,y_3]_c=0;\\
y_{3}^2&=\mu_3; & y_{443}&=0; & [y_{(24)},&y_3]_c=0; & &[y_{236},y_3]_c=0.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{34} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $A_4$ and $A_2$.

\subsubsection{Type $\mathfrak{g}(8,6)$, $\zeta \in \mathrm{G}'_3$}\label{subsec:type-g(8,6)}
The Weyl groupoid has eight objects.

 For $\xymatrix{\overset{\zeta}{\underset{\ }{\circ}}\ar  @{-}[r]^{\overline{\zeta}}  &
\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
}{\circ}}
\ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}}
\ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\ }{\circ}}  \ar  @{-}[r]^{\overline{\zeta}}  &
\overset{\zeta}{\underset{\ }{\circ}}  \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
}{\circ}}
}$, $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_7$, with defining relations
\begin{align*}
y_{5}^2&=\mu_5; & y_{112}&=\lambda_{112}; & y_{221}&=\lambda_{221}; & y_{223}&=\lambda_{223}; \\
y_{665}&=0; & y_{667}&=\lambda_{667}; & y_{332}&=\lambda_{332}; & y_{334}&=\lambda_{334};\\
& & y_{443}&=\lambda_{443}; & y_{445}&=\lambda_{445}; & y_{776}&=\lambda_{776};
\end{align*}
\vspace*{-0.7cm}
\begin{align*}
y_{ij}&=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1; & [[[y_{(36)},y_5]_c,y_4]_c,y_5]_c&=[[[y_{7654},y_5]_c,y_6]_c,y_5]_c=0.
\end{align*}

Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{56} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $A_5$ and $A_2$.

 For $\xymatrix@R-8pt{  &   & & \overset{-1}{\circ} \ar  @{-}[d]^{\zeta}\ar  @{-}[dr]^{\zeta}
& &\\
\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}
 & \overset{-1}{\underset{\ }{\circ}}
\ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  &
\overset{\zeta}{\underset{\ }{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_7$, with defining relations
\begin{align*}
y_{4}^2&=\mu_4; & y_{112}&=\lambda_{112}; & y_{221}&=\lambda_{221}; & y_{ij}&=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{5}^2&=\mu_5; & y_{223}&=\lambda_{223}; & y_{332}&=\lambda_{332}; & [y_{(35)},&y_4]_c=[y_{347},y_4]_c=0;\\
y_{7}^2&=\mu_7; & y_{334}&=0; & y_{665}&=0; & [y_{(46)},&y_5]_c=[y_{657},y_5]_c=0
\end{align*}
\vspace*{-0.7cm}
\begin{align*}
y_{457}-q_{57}\overline{\zeta} [y_{47},y_5]_c-q_{45}(1-\zeta)y_5y_{47}=0.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{47},y_{57} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $A_6$ and $A_1$.

 For $\xymatrix@R-8pt{  &  & & & \overset{\zeta}{\circ} \ar  @{-}[d]^{\overline{\zeta}}  & \\
\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}
 & \overset{-1}{\underset{\ }{\circ}}
\ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  &
\overset{-1}{\underset{\ }{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_7$, with defining relations
\begin{align*}
y_{4}^2&=\mu_4; & y_{112}&=\lambda_{112}; & y_{221}&=\lambda_{221}; &  y_{ij}&=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{5}^2&=\mu_5; & y_{223}&=\lambda_{223}; & y_{332}&=\lambda_{332}; & [y_{(35)},&y_4]_c=[[y_{65},y_{657}]_c,y_5]_c=0;\\
y_{6}^2&=\mu_6; & y_{775}&=0; & [y_{(46)},&y_5]_c=\mu_5; & y_{334}&=[y_{457},y_5]_c=0.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{57} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $A_6$ and $A_1$.

 For $\xymatrix@R-8pt{  &  & & & \overset{\zeta}{\circ} \ar  @{-}[d]^{\overline{\zeta}}  & \\
\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}
 & \overset{\zeta}{\underset{\ }{\circ}}
\ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  &
\overset{-1}{\underset{\ }{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_7$, with defining relations
\begin{align*}
y_{112}&=\lambda_{112}; & y_{221}&=\lambda_{221}; & y_{223}&=\lambda_{223}; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{557}&=\lambda_{557}; & y_{332}&=\lambda_{332}; & y_{334}&=\lambda_{334}; &  &y_{443}=\lambda_{443};\\
y_{775}&=\lambda_{775}; & y_{445}&=\lambda_{445}; & y_{554}&=\lambda_{554}; & &y_{556}=0; \qquad y_{6}^2=\mu_2.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{56} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $A_6$ and $A_1$.

 For $\xymatrix@R-8pt{  &   & & \overset{\zeta}{\circ} \ar  @{-}[d]^{\overline{\zeta}}  & &\\
\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}
& \overset{-1}{\underset{\ }{\circ}}
\ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  &
\overset{\zeta}{\underset{\ }{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_7$, with defining relations
\begin{align*}
y_{112}&=\lambda_{112}; & y_{221}&=\lambda_{221}; & [y_{347},&y_4]_c=0; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{3}^2&=\mu_3; & y_{556}&=\lambda_{556}; & [y_{(24)},&y_3]_c=0; &  &[[[y_{6547},y_4]_c,y_5]_c,y_4]_c=0;\\
y_{4}^2&=\mu_4; & y_{665}&=\lambda_{665}; & [y_{(35)},&y_4]_c=0; & &y_{223}=y_{774}=y_{554}=0.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{47} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $A_6$ and $A_1$.

For $\xymatrix@R-8pt{  &   & & \overset{\zeta}{\circ} \ar  @{-}[d]^{\overline{\zeta}}  & & \\
\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\
}{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}
& \overset{\zeta}{\underset{\ }{\circ}}
\ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  &
\overset{\zeta}{\underset{\ }{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_7$, with defining relations
\begin{align*}
y_{112}&=y_{443}=0; & y_{665}&=\lambda_{665}; & y_{445}&=\lambda_{445}; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{2}^2&=\mu_2; & y_{447}&=\lambda_{447}; & y_{774}&=\lambda_{774}; & &[y_{(13)},y_2]_c=0;\\
y_{3}^2&=\mu_3; & y_{554}&=\lambda_{554}; & y_{556}&=\lambda_{556}; & &[y_{(24)},y_3]_c=0.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{34} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $A_3$ and $A_4$.

 For $\xymatrix@R-8pt{  &   & & \overset{\zeta}{\circ} \ar  @{-}[d]^{\overline{\zeta}}    & &\\
\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\
}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}
 & \overset{\zeta}{\underset{\ }{\circ}}
\ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  &
\overset{\zeta}{\underset{\ }{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_7$, with defining relations
\begin{align*}
y_{1}^2&=\mu_1; & y_{334}&=\lambda_{334}; & y_{443}&=\lambda_{443}; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{2}^2&=\mu_2; & y_{445}&=\lambda_{445}; & y_{447}&=\lambda_{447}; & &[y_{(13)},y_2]_c=0;\\
y_{665}&=\lambda_{665} & y_{774}&=\lambda_{774}; & y_{554}&=\lambda_{554}; & & y_{556}=\lambda_{556}; \qquad y_{332}=0.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{23} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $A_2$ and $D_5$.

 For $\xymatrix@R-8pt{  &   & & \overset{\zeta}{\circ} \ar  @{-}[d]^{\overline{\zeta}}    & &\\
\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}
 & \overset{\zeta}{\underset{\ }{\circ}}
\ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  &
\overset{\zeta}{\underset{\ }{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_7$, with defining relations
\begin{align*}
y_{221}&=0; & y_{223}&=\lambda_{223}; & y_{332}&=\lambda_{332}; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{556}&=\lambda_{556}; & y_{334}&=\lambda_{334}; & y_{443}&=\lambda_{443}; & &y_{445}=\lambda_{445};\\
y_{665}&=\lambda_{665}; & y_{447}&=\lambda_{447}; & y_{774}&=\lambda_{774}; & &y_{554}=\lambda_{554}; \qquad y_{1}^2=\mu_1.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{12} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $A_1$ and $E_6$.

\subsection{Super modular type, characteristic
5}\label{sec:by-diagram-super-modular-char5}
\subsubsection{Type $\mathfrak{brj}(2; 5)$, $\zeta \in \mathrm{G}'_5$}\label{subsec:type-brj(2;5)}
The Weyl groupoid has two objects.

For $\xymatrix{\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta^2} &
\overset{-1}{\underset{\ }{\circ}}}$, $\widetilde{\mathcal E}_{\bq}$ is generated by $y_1, y_2$ with defining relations
\begin{align*}
[y_{1112},y_{112}]_c&=\lambda_1;  &
 y_2^2&=\lambda_2;  & y_{11112}&=0; &  [[[y_{112},y_{12}]_c,y_{12}]_c,y_{12}]_c&=0,
\end{align*}
with $\lambda_1\neq0$ only if ${\mathfrak q}=\left(\begin{smallmatrix}
\zeta & 1 \\\zeta^2 &-1
\end{smallmatrix}\right)$. See \texttt{brj25.log}, resp. \texttt{brj252.log}, for the deformation of the first, resp. last, relation.
If $\lambda_1=0$, then this algebra is nonzero by \cite[Lemma 5.16]{AAGMV}. If $\lambda_1,\lambda_2\neq0$, then this algebra is nonzero by
 \texttt{brj25b.log}.

For $\xymatrix{\overset{-\zeta^3}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta^3} &
\overset{-1}{\underset{\ }{\circ}}}$, $\widetilde{\mathcal E}_{\bq}$ is generated by $y_1, y_2$ with defining relations
\begin{align*}
 y_2^2&=\lambda_2;  & y_{111112}&=0; & [y_1,&[y_{112},y_{12}]_c]_c+q_{12}y_{112}^2=0.
\end{align*}
This algebra is nonzero by \cite[Lemma 5.16]{AAGMV}.

\subsubsection{Type $\mathfrak{el}(5;5)$, $\zeta \in \mathrm{G}'_{5}$}\label{subsec:type-el(5;5)}
The Weyl groupoid has seven objects.

 For $\xymatrix@C-4pt{\overset{\zeta^2}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}^{\, 2}} &
\overset{\zeta^{2}}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}^{\, 2}}  &
\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
}{\circ}} \ar  @{-}[r]^{\overline{\zeta}^{\, 2}}  & \overset{\zeta^2}{\underset{\ }{\circ}}}$
the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
\begin{align*}
y_{112}&=0; & y_{221}&=0; & y_{223}&=0; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{554}&=\lambda_{554}; & y_{443}&=0; & y_{4445}&=\lambda_{4445}; &  &[[[y_{(14)},y_3]_c,y_2]_c,y_3]_c=0;
\end{align*}
\vspace*{-0.7cm}
\begin{align*}
y_{3}^2&=\mu_3; & [[y_{5432},y_4]_c,y_3]_c-&q_{43}(\zeta^2-\zeta)[[y_{5432},y_3]_c,y_4]_c=0.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{34} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $A_3$ and $B_2$.

 For $\xymatrix@C-4pt{\overset{\zeta^2}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}^{\, 2}} &
\overset{\zeta^{2}}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}^{\, 2}}  &
\overset{\zeta^{2}}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}^{\, 2}}  &
\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\
}{\circ}}}$
the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
\begin{align*}
y_{112}&=0; & y_{4}^2&=\mu_4; & y_{221}&=0; & y_{223}&=0; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
& & y_{5}^2&=\mu_5; & y_{332}&=0; & y_{334}&=0; &  &[[y_{54},y_{543}]_c,y_4]_c=0.
\end{align*}
This algebra is nonzero by \cite[Lemma 5.16]{AAGMV}.

 For $\xymatrix@C-4pt{\overset{\zeta^2}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}^{\, 2}} &
\overset{\zeta^{2}}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}^{\, 2}}  &
\overset{\zeta^{2}}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}^{\, 2}}  &
\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\
}{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
\begin{align*}
y_{5}^2&=\mu_5; & y_{112}&=0; & y_{221}&=0; & y_{223}&=0; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
& & y_{445}&=0; & y_{332}&=0; & y_{334}&=\lambda_{334}; & &y_{4443}=\lambda_{4443}.
\end{align*}
Here, $\widetilde{\mathcal E}_{\bq}/ \langle y_{45} \rangle \simeq \widetilde{\mathcal E}_{{\mathfrak q}'}$, ${\mathfrak q}'$ with two components of types $A_1$ and $B_4$.

 For $\xymatrix@R-8pt{  &   \overset{\zeta^2}{\circ} \ar  @{-}[d]^{\overline{\zeta}^{\, 2}} & & \\
\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta^2}  & \overset{-1}{\underset{\
}{\circ}} \ar  @{-}[r]^{\overline{\zeta}^{\, 2}}  & \overset{\zeta^2}{\underset{\ }{\circ}} \ar
@{-}[r]^{\overline{\zeta}^{\, 2}}  & \overset{\zeta^2}{\underset{\ }{\circ}}}$
the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
\begin{align*}
y_{332}&=0; & y_{1}^2&=\mu_1; & y_{334}&=0; & y_{443}&=0; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
& & y_{2}^2&=\mu_2; & y_{552}&=0; & [y_{(13)},&y_2]_c=0; & &[y_{125},y_2]_c=0.
\end{align*}
This algebra is nonzero by \cite[Lemma 5.16]{AAGMV}.

For $\xymatrix@R-8pt{  &   \overset{\zeta^2}{\circ} \ar  @{-}[d]^{\overline{\zeta}^{\, 2}} & & \\
\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}^{\, 2}}  &
\overset{\zeta^2}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}^{\, 2}}  &
\overset{\zeta^2}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}^{\, 2}}  &
\overset{\zeta^2}{\underset{\ }{\circ}}}$
the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
\begin{align*}
y_{1}^2&=\mu_1; & y_{ij}&=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1; & y_{iij}&=0,  \, \widetilde{q}_{ij}\neq 1, \, i\neq 1.
\end{align*}
This algebra is nonzero by \cite[Lemma 5.16]{AAGMV}.

 For $\xymatrix@R-8pt{  &   \overset{-1}{\circ} \ar  @{-}[d]_{\zeta^{2}} \ar  @{-}[dr]^{\zeta}
& & \\
\overset{\zeta^2}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}^{\, 2}}  &
\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta^{2}}  & \overset{-1}{\underset{\
}{\circ}} \ar  @{-}[r]^{\overline{\zeta}^{\, 2}}  & \overset{\zeta^2}{\underset{\ }{\circ}}}$
the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
\begin{align*}
y_{112}&=0; & y_{443}&=0; & [y_{(13)},&y_2]_c=0; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
[y_{125},&y_2]_c=0; & y_{2}^2&=\mu_2; & [y_{(24)},&y_3]_c=0; & &[[y_{53},y_{534}]_c,y_3]_c=0;
\end{align*}
\vspace*{-0.7cm}
\begin{align*}
y_{3}^2&=\mu_3; & y_{5}^2&=\mu_5; & y_{235}-\frac{q_{35}}{\zeta^2+\zeta}[y_{25},y_3]_c-q_{23}(1-\zeta)y_3y_{25}=0.
\end{align*}
This algebra is nonzero by \cite[Lemma 5.16]{AAGMV}.

 For $\xymatrix@R-8pt{& &  \overset{\zeta}{\circ} \ar  @{-}[d]_{\overline{\zeta}} \ar  @{-}[dr]^{\overline{\zeta}} &
\\
\overset{\zeta^{2}}{\underset{\ }{\circ}}  \ar  @{-}[r]^{\overline{\zeta}^{\, 2}} &
\overset{\zeta^{2}}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}^{\, 2}}  &
\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta^{2}}  & \overset{-1}{\underset{\
}{\circ}} }$
the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
\begin{align*}
y_{112}&=0; & y_{221}&=0; & y_{223}&=0; & & y_{ij}=\lambda_{ij}, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{554}&=0; & y_{553}&=0; & [y_{(24)},&y_3]_c=0; &  &[[[y_{1235},y_3]_c,y_2]_c,y_3]_c=0;
\end{align*}
\vspace*{-0.7cm}
\begin{align*}
y_{3}^2&=\mu_3; & y_4^2&=\mu_4; & y_{(35)}-q_{45}\zeta[y_{35},y_4]_c-q_{34}(1-\overline{\zeta})y_4y_{35}=0.
\end{align*}
This algebra is nonzero by \cite[Lemma 5.16]{AAGMV}.

\subsection{Unidentified type}\label{sec:by-diagram-Unidentified}

\subsubsection{Type $\mathfrak{wk}(4,\alpha)$, $q\neq \pm 1$}\label{subsec:type-bgl(4,alpha)}
The Weyl groupoid has five objects.
Two of them are obtained with $-q^{-1}$
instead of $q$; hence we study the other three.
We set $N=\ord q$, $M=\ord -q^{-1}$.

For $\xymatrix@C-4pt{\overset{q}{\underset{\ }{\circ}}\ar  @{-}[r]^{q ^{-1}}  &
\overset{q}{\underset{\ }{\circ}}
\ar  @{-}[r]^{q^{-1}}  & \overset{-1}{\underset{\ }{\circ}}
\ar  @{-}[r]^{-q}  & \overset{-q^{-1}}{\underset{\ }{\circ}}}$, the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining relations
\begin{align*}
 y_3^2&=\lambda_3; & y_{13}&=0; & y_{14}&=0; & y_{24}&=0; \\
y_{112}&=\lambda_1; & y_{221}&=\lambda_2; & y_{223}&=\lambda_4; &  y_{443}&=\lambda_5.
\end{align*}
Here $\lambda_1\lambda_2\neq 0$ or $\lambda_4\neq 0$ only if $N=3$. Also, $\lambda_1\lambda_4=\lambda_2\lambda_4=0$. Similarly, $\lambda_5\neq 0$ only if $M=4$. In any case we can project onto a case of smaller rank and hence $\widetilde{\mathcal E}_{\bq}\neq0$.

For $\xymatrix@C-4pt{\overset{q}{\underset{\ }{\circ}}\ar  @{-}[r]^{q ^{-1}}  &
\overset{-1}{\underset{\ }{\circ}}
\ar  @{-}[r]^{-1}  & \overset{-1}{\underset{\ }{\circ}}
\ar  @{-}[r]^{-q}  & \overset{-q^{-1}}{\underset{\ }{\circ}}}$, the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining relations
\begin{align*}
 y_2^2&=\lambda_2; & y_{13}&=0; & y_{14}&=0; & y_{24}&=0; \\
y_{3}^2&=\lambda_3; & y_{23}^2&=\lambda_4; & y_{112}&=\lambda_1; &  y_{443}&=\lambda_5;
\end{align*}
\vspace*{-0.7cm}
\begin{align*}
[[y_{(14)},y_2 ]_c,y_3]_c-q_{23}\frac{1+q}{1-q}[[ y_{(14)},y_3]_c,y_2]_c=0.
\end{align*}
Here, $\lambda_1\lambda_5=0$  and $\lambda_1$ or $\lambda_5$ are nonzero only if $N=4$. Thus, we can always project onto a case of smaller rank and hence $\widetilde{\mathcal E}_{\bq}\neq 0$. See \texttt{bgla.log}, resp. \texttt{bgla1.log}, \texttt{bgla2.log} for the deformation of the longest relation when one, resp. two, resp. three, of the relations $y_2^2,y_3^2,y_{23}^2$ are deformed.

For $\xymatrix@C-5pt{ \overset{q}{\underset{\ }{\circ}} \ar  @{-}[r]^{q
^{-1}}  & \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{-1}
\ar@/^2pc/  @{-}[rr]^{q}
 & \overset{-1}{\underset{\ }{\circ}}
\ar  @{-}[r]^{-q^{-1}} & \overset{-1}{\circ}}$, the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining relations
\begin{align*}
 y_{112}&=\lambda_1; & y_{13}&=0; & y_{14}&=\lambda_5; & [y_{(13)},&y_2]_c=0; \\
 y_{2}^2&=\lambda_2; & y_{24}^2&=\lambda_6; & y_{3}^2&=\lambda_3; &  y_{4}^2&=\lambda_4;
\end{align*}
\vspace*{-0.7cm}
\begin{align*}
y_{(24)}+\frac{(1+q)q_{43}}{2}[y_{24},y_3]_c-q_{23}(1+q^{-1})y_3y_{24}=0.
\end{align*}
Here $\lambda_1\neq 0$ only if $N=4$. Hence, if $\lambda_3=0$ we can project onto a case of smaller rank. If $\lambda_3\neq 0$, then $\lambda_4=\lambda_5=\lambda_6=0$ and we can project onto a case of smaller rank. Assume $\lambda_1=0$. If $\lambda_5\neq0$, then we can project. Otherwise, $\lambda_2=\lambda_4=0$ and we have that either $\lambda_6=0$ or $\lambda_3=0$.
In any case, we can project. Hence $\widetilde{\mathcal E}_{\bq}\neq0$.

\subsubsection{Type $\mathfrak{ufo}(1)$, $\zeta\in\mathrm{G}_4$}\label{subsec:type-ufo(1)}
The Weyl groupoid has six objects.

For $\Dchainfive{\zeta }{\overline{\zeta} }{\zeta }{\overline{\zeta} }{-1}{-1}{-1}{\overline{\zeta} }{\zeta }$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by
$y_i$, $i\in\I_5$, with defining relations
\begin{align*}
y_3^2&=\lambda_3; & y_{112}&=0; & y_{221}&=0; & y_{ij}&=0, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_4^2&=\lambda_4; & y_{223}&=\lambda_1; & y_{554}&=\lambda_2; &  [[[y_{(14)},&y_3]_c,y_2]_c,y_3]_c=0;
\end{align*}
\vspace*{-0.7cm}
\begin{align*}
y_{34}^2&=\lambda_5; & [[y_{(25)},y_3]_c,y_4]_c-q_{34}\zeta[[y_{(25)},y_4]_c,y_3]_c=0.
\end{align*}
Here $\widetilde{\mathcal E}_{\bq}\neq 0$ as we can project onto a case of smaller rank by making $y_1=0$.

For $\Dchainfive{\zeta }{\overline{\zeta} }{\zeta }{\overline{\zeta} }{\zeta }{\overline{\zeta} }{-1}{\overline{\zeta} }{\zeta }$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
\begin{align*}
 y_{112}&=0; & y_{221}&=0;  & y_{4}^2&=\lambda_1 & y_{ij}&=0, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{223}&=0; & y_{332}&=0; &  y_{334}&=\lambda_2; & y_{554}&=\lambda_3.
 \end{align*}
In this case $\widetilde{\mathcal E}_{\bq}\neq 0$ as we can project onto a case of smaller rank.

For $\xymatrix@R-8pt{  &   \overset{-1}{\circ} \ar  @{-}[d]_{\zeta} \ar  @{-}[dr]^{-1} & & \\
\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\
}{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}
 & \overset{\zeta}{\underset{\ }{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
\begin{align*}
y_{2}^2&=\lambda_2; & y_3^2&=\lambda_3;& y_5^2&=\lambda_5; & [y_{(13)},&y_2]_c=0;  & y_{ij}&=0, \, i<j, \, \widetilde{q}_{ij}=1;\\
&& y_{112}&=\lambda_1; & y_{443}&=\lambda_4; &  [y_{125},&y_2]_c=0; &  [y_{(24)},& y_3]_c=0;
 \end{align*}
\vspace*{-0.7cm}
\begin{align*}
y_{35}^2&=\lambda_6; & y_{235}+q_{35}(1+\zeta)[y_{25},y_3]_c-2q_{23}y_3y_{25}=0.
\end{align*}
If $\lambda_1=0$, then we can project. If $\lambda_1\lambda_3\neq 0$, then $\lambda_4=0$, and we can make $y_4=0$.
If $\lambda_1\lambda_4\neq 0$, then $\lambda_5=0$, and we can make $y_5=0$. Hence $\widetilde{\mathcal E}_{\bq}\neq 0$.

For $\xymatrix@R-8pt{& &  \overset{-1}{\circ} \ar  @{-}[d]_{-1} \ar  @{-}[dr]^{\zeta} &  \\
\overset{\zeta}{\underset{\ }{\circ}}  \ar  @{-}[r]^{\overline{\zeta}} &
\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\
}{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\ }{\circ}} }$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
\begin{align*}
 y_{112}&=0; & y_{221}&=0; & y_{223}&=\lambda_1; & y_{ij}&=0, && i<j, \widetilde{q}_{ij}=1;\\
y_5^2&=\lambda_2; & y_{35}^2&=\lambda_5; & y_{3}^2&=\lambda_3; & y_4^2&=\lambda_4; &&  [[[y_{1235},y_3]_c,y_2]_c,y_3]_c=0;
 \end{align*}
\vspace*{-0.7cm}
\begin{align*}
 [y_{(24)},&y_3]_c=0; & y_{(35)}+\frac{q_{45}(1+\zeta)}{2}[y_{35},y_4]_c-q_{34}(1-\zeta)y_4y_{35}=0.
\end{align*}
Hence $\widetilde{\mathcal E}_{\bq}\neq 0$ as we can quotient by $y_{34}$, $y_{45}$.

For $\xymatrix@R-8pt{  &   \overset{\zeta}{\circ} \ar  @{-}[d]^{\overline{\zeta}} & & \\
\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\
}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}} \ar
@{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
\begin{align*}
y_1^2&=\lambda_1; & y_2^2&=\lambda_2 & [y_{(13)},&y_2]_c=0; & y_{ij}&=0, &&\quad i<j, \widetilde{q}_{ij}=1;\\
 y_{332}&=\lambda_3; & y_{334}&=0; & y_{443}&=0; & y_{552}&=\lambda_4; &&  [y_{125},y_2]_c=0.
\end{align*}
We can project by making $y_4=0$, hence $\widetilde{\mathcal E}_{\bq}\neq 0$.

For $\xymatrix@R-8pt{  &   \overset{\zeta}{\circ} \ar  @{-}[d]^{\overline{\zeta}} & & \\
\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}} \ar
@{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_5$, with defining relations
\begin{align*}
 y_{221}&=\lambda_1; & y_{223}&=0; & y_{225}&=0; & y_{ij}&=0, & &  i<j, \widetilde{q}_{ij}=1;\\
y_1^2&=\lambda_2; & y_{552}&=0; & y_{332}&=0; & y_{334}&=0;&  & y_{443}=0.
\end{align*}
We can project onto a case of smaller rank, hence $\widetilde{\mathcal E}_{\bq}\neq 0$.

\subsubsection{Type $\mathfrak{ufo}(2)$, $\zeta \in \mathrm{G}'_4$}\label{subsec:type-ufo(2)}
The Weyl groupoid has seven objects.

For $\xymatrix@C-4pt{\overset{\zeta}{\underset{\ }{\circ}}\ar  @{-}[r]^{\overline{\zeta}}  &
\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
}{\circ}}
\ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{-1}  &
\overset{-1}{\underset{\ }{\circ}}  \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
}{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_6$, with defining relations
\begin{align*}
y_{112}&=y_{221}=0; & y_{334}&=\lambda_1; & y_4^2&=\lambda_3; & y_{ij}&=0, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{332}&=y_{223}=0; & y_{665}&=\lambda_2; & y_5^2&=\lambda_4; & [[[y_{(25)},&y_4]_c,y_3]_c,y_4]_c=0;
\end{align*}
\vspace*{-0.7cm}
\begin{align*}
y_{45}^2&=\lambda_5; & [[y_{(36)},y_4]_c,y_5]_c-&q_{45}\zeta[[y_{(36)},y_5]_c,y_4]_c=0.
\end{align*}
We can project onto a case of smaller rank, hence $\widetilde{\mathcal E}_{\bq}\neq 0$.

For $\xymatrix@C-4pt{\overset{\zeta}{\underset{\ }{\circ}}\ar  @{-}[r]^{\overline{\zeta}}  &
\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
}{\circ}}
\ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  &
\overset{-1}{\underset{\ }{\circ}}  \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
}{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_6$, with defining relations
\begin{align*}
y_{445}&=\lambda_1; & y_{112}&=0; & y_{221}&=0; & y_{223}&=0; & y_{ij}&=0, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{665}&=\lambda_2; & y_{332}&=0; &  y_{334}&=0; & y_{443}&=0; & y_{5}^2&=\lambda_3
\end{align*}
We can project by making $y_1=y_2=y_3=0$, hence $\widetilde{\mathcal E}_{\bq}\neq 0$.

For $\xymatrix@C-5pt@R-8pt{  &   &  \overset{\zeta}{\circ} \ar  @{-}[d]^{\overline{\zeta}} & & \\
\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\
}{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}
& \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
}{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_6$, with defining relations
\begin{align*}
y_{112}&=\lambda_1; & y_{443}&=\lambda_2; & y_2^2&=\lambda_4; & y_{ij}&=0, \, i<j, \, \widetilde{q}_{ij}=1; \quad y_{554}=0\\
y_{445}&=0; & y_{663}&=\lambda_3; & y_3^2&=\lambda_5; & [y_{(13)},&y_2]_c= [y_{(24)},y_3]_c= [y_{236},y_3]_c=0.
\end{align*}
Hence $\widetilde{\mathcal E}_{\bq}\neq 0$ as we can project onto smaller rank.

For $\xymatrix@C-5pt@R-8pt{  &   & \overset{\zeta}{\circ} \ar  @{-}[d]^{\overline{\zeta}}  & &\\
\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\
}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}
 & \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  &
\overset{\zeta}{\underset{\ }{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_6$, with defining relations
\begin{align*}
y_1^2&=\lambda_1; & y_{332}&=\lambda_2; & y_{336}&=y_{663}=0; & y_{ij}&=0, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_2^2&=\lambda_3; & [y_{(13)},&y_2]_c=0; & y_{334}&=y_{443}=0; & y_{445}&=y_{554}=0.
\end{align*}
We can project onto a case of smaller rank, hence $\widetilde{\mathcal E}_{\bq}\neq 0$.

For $\xymatrix@C-5pt@R-8pt{  &   & & \overset{-1}{\circ} \ar  @{-}[d]^{-1} \ar
@{-}[dr]^{\zeta}  & \\
\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}
 & \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}  & \overset{-1}{\underset{\
}{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_6$, with defining relations
\begin{align*}
y_4^2&=\lambda_1; & y_5^2&=\lambda_2; &  [y_{(35)},&y_4]_c=0; & y_{223}&=0; & &y_{ij}=0, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_6^2&=\lambda_3; & y_{46}^2&=\lambda_4; & y_{332}&=0; & y_{334}&=\lambda_5; &  &[[[y_{2346},y_4]_c,y_3]_c,y_4]_c=0;
\end{align*}
\vspace*{-0.7cm}
\begin{align*}
y_{112}&=y_{221}=0; & y_{(46)}+\frac{q_{56}(1+\zeta)}{2}[y_{46},y_5]_c-q_{45}(1-\zeta)y_5y_{46}=0.
\end{align*}
We can project onto smaller rank by making $y_1=y_2=0$, hence $\widetilde{\mathcal E}_{\bq}\neq 0$.

For $\xymatrix@C-5pt@R-8pt{  &   & \overset{\zeta}{\circ} \ar  @{-}[d]^{\overline{\zeta}}  & &\\
\overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}
 & \overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  &
\overset{\zeta}{\underset{\ }{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_6$, with defining relations
\begin{align*}
y_{223}&=y_{332}=0; & y_{663}&=y_{336}=0; & y_{ij}&=0, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{334}&=y_{443}=0; & y_{445}&=y_{554}=0; & y_1^2&=\lambda_1; \quad y_{221}=\lambda_2.
\end{align*}
We can project onto a case of smaller rank, hence $\widetilde{\mathcal E}_{\bq}\neq 0$.

For $\xymatrix@C-5pt@R-8pt{  &   & \overset{-1}{\circ} \ar  @{-}[d]^{\zeta} \ar
@{-}[dr]^{-1}  & &\\
\overset{\zeta}{\underset{\ }{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\
}{\circ}} \ar  @{-}[r]^{\overline{\zeta}}  & \overset{-1}{\underset{\ }{\circ}} \ar  @{-}[r]^{\zeta}
& \overset{-1}{\underset{\ }{\circ}}
\ar  @{-}[r]^{\overline{\zeta}}  & \overset{\zeta}{\underset{\ }{\circ}}}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_6$, with defining relations
\begin{align*}
y_{3}^2&=\lambda_1; & y_4^2&=\lambda_2; &  y_{223}&=\lambda_3; & [y_{(24)},&y_3]_c=0; & y_{ij}&=0, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_6^2&=\lambda_4; & y_{46}^2&=\lambda_5; & y_{554}&=\lambda_6; & [y_{(35)},&y_4]_c=0; & y_{112}&=y_{221}=0;
\end{align*}
\vspace*{-0.7cm}
\begin{align*}
[y_{236},&y_3]_c=0; & y_{346}+q_{46}(1+\zeta)&[y_{36},y_4]_c-2q_{34}y_4y_{36}=0.
\end{align*}
We can project onto smaller rank by making $y_1=y_2=0$, hence $\widetilde{\mathcal E}_{\bq}\neq 0$.

\subsubsection{Type $\mathfrak{ufo}(3)$, $\zeta \in \mathrm{G}'_3$}\label{subsec:type-ufo(3)}
The Weyl groupoid has five objects.

For $\Dchainthree{-1}{\overline{\zeta}}{\zeta }{-\overline{\zeta}}{-\zeta }$, the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_3$, with defining relations
\begin{align*}
 y_1^2&=\lambda_1; & y_2^3&=\lambda_2; & y_{13}&=0; & y_{221}&=0; &  y_{332}&=\lambda_3,
\end{align*}
with $\lambda_1\lambda_3=\lambda_2\lambda_3=0$. Hence $\widetilde{\mathcal E}_{\bq}$ projects onto a nonzero rank-2 case.

For $\Dchainthree{-1}{\zeta }{-1}{-\overline{\zeta}}{-\zeta }$, the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_3$, with defining relations
\begin{align*}
 y_{13}&=0; & y_{332}&=0; &  y_1^2&=\lambda_1; & y_2^2&=\lambda_2.
\end{align*}
This algebra is nonzero by \cite[Lemma 5.16]{AAGMV}.

For $\Dchainthree{\zeta }{-1}{-1}{-\overline{\zeta}}{-\zeta }$, the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_3$, with defining relations
\begin{align*}
y_1^3&=\lambda_1; & y_2^2&=\lambda_2; & [y_{112},y_{12}]_c&=\lambda_3;  &  [[y_{12},&y_{(13)}]_c,y_2]_c= y_{13}=y_{332}=0.
\end{align*}
Here $\widetilde{\mathcal E}_{\bq}$ projects onto a rank 2 case by making $y_3=0$.

For $\Dchainthree{\zeta }{-\overline{\zeta}}{-\zeta }{-\overline{\zeta}}{-\zeta }$, the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_3$, with defining relations
\begin{align*}
 y_{13}&=0; & y_{221}&=\lambda_2; &  y_1^3&=\lambda_1; & y_{223}&=0; & y_{332}&=0.
\end{align*}
This algebra is nonzero by \cite[Lemma 5.16]{AAGMV}.

For $\Dtriangle{\zeta }{-1}{-1}{\overline{\zeta}}{-\zeta }{-1}$, the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_3$, with defining relations
\begin{align*}
 y_{113}&=0; & y_1^3&=\lambda_1; &  [y_{112},y_{12}]_c&=\lambda_3; & y_2^2&=\lambda_2; &  y_3^2&=\lambda_4
\end{align*}
\vspace*{-0.7cm}
\begin{align*}
y_{(13)}-\frac{q_{23}\zeta}{1-\overline{\zeta}}[y_{13},y_2]_c+q_{12}\overline{\zeta} y_2y_{13}=0,
\end{align*}
with $\lambda_2\lambda_4=0$. Hence the algebra $\widetilde{\mathcal E}_{\bq}$ projects onto a rank-2 nonzero algebra by setting $y_2=0$ or $y_3=0$ according to whether $\lambda_4=0$ or $\lambda_2=0$.

\subsubsection{Type $\mathfrak{ufo}(4)$, $\zeta \in \mathrm{G}'_3$}\label{subsec:type-ufo(4)}
The Weyl groupoid has nine objects.

For $\Dchainthree{-1}{-1}{-1}{\zeta }{-1}$, the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_3$, with defining relations
\begin{align*}
 y_1^2&=\lambda_1; & y_2^2&=\lambda_2; & y_3^2&=\lambda_3; &  y_{23}^3&=\lambda_6;\\
& & y_{12}^2&=\lambda_4; & y_{13}&=\lambda_5; & [y_{123},[y_{123},y_{23}]_c]_c&=\lambda_7
\end{align*}
with $\lambda_2\lambda_3=\lambda_7\lambda_5=0$. The deformation of $y_{23}^3$ follows from \texttt{ufo4a1.log} and of the longest relation by \texttt{ufo4a2.log}.

Assume $\lambda_7=0$. Then we can project onto a case of smaller rank unless $\lambda_1\lambda_2\lambda_4\lambda_6\neq 0$ or $\lambda_1\lambda_3\lambda_6\neq 0$, which gives
${\mathfrak q}=
\left(\begin{smallmatrix}
-1& \pm 1 & \pm 1\\ \mp1 & -1 & -\zeta \\ \pm 1 &-1 &-1
\end{smallmatrix}\right)
$ or ${\mathfrak q}=
\left(\begin{smallmatrix}
-1& \pm 1 & \pm 1\\ \mp1 & -1 & -1 \\ \pm 1 &-\zeta &-1
\end{smallmatrix}\right)
$. In any case, $\widetilde{\mathcal E}_{\bq}\neq 0$, see \texttt{ufo4a.log}.

Now $\lambda_7\neq 0$ only if
${\mathfrak q}=
\left(\begin{smallmatrix}
-1& a & a^{-1}\\ -a^{-1} & -1 & c\\ a & c^{-1}\zeta &-1
\end{smallmatrix}\right)$, with $c^3=-a^2$. In any case, $\widetilde{\mathcal E}_{\bq}\neq 0$, see \texttt{ufo4a7.log}.

For $\Dchainthree{-1}{-1}{-1}{-\overline{\zeta}}{\overline{\zeta}}$, the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_3$, with defining relations
\begin{align*}
 y_1^2&=\lambda_1; & y_2^2&=\lambda_2; & y_3^3&=\lambda_3 & [y_{332},y_{32}]_c&=\lambda_4;  \\
 [y_{3321},y_{321}]_c&=\lambda_5; & y_{13}&=0; &  y_{12}^2&=\lambda_6; & [[y_{32},y_{321}]_c,y_2]_c&=0.
\end{align*}
We can project onto a case of rank two unless ${\mathfrak q}=
\left(\begin{smallmatrix}
-1& \pm 1 & 1 \\ \mp 1 & -1 & \zeta^{-1}\\ 1 & -1 &\zeta^{-1}
\end{smallmatrix}\right)
$, when all scalars can be nonzero. In this case, $\widetilde{\mathcal E}_{\bq}\neq0$ by \texttt{ufo4b.log}.

For $\Dchainthree{-1}{-1}{\zeta }{\overline{\zeta}}{-1}$, the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_3$, with defining relations
\begin{align*}
 y_1^2&=\lambda_1; & y_2^3&=\lambda_2; & y_3^2&=\lambda_3; & [y_{221}, y_{21}]_c&=\lambda_5; \\
y_{223}&=0; & y_{13}&=\lambda_4;  & [y_{12},y_{(13)}]_c&=0,
\end{align*}
with $\lambda_1\lambda_2=0=\lambda_2\lambda_3=\lambda_2\lambda_5=\lambda_2\lambda_4$. Hence, if $\lambda_2\neq0$, then $\widetilde{\mathcal E}_{\bq}$ projects onto $\Bbbk\langle y_2| y_2^3=\lambda_2\rangle$. If $\lambda_1\neq0$, then $\lambda_5=0$ and thus $\widetilde{\mathcal E}_{\bq}$ projects onto the linking of two rank-one algebras, by making $y_2=0$.
A similar situation holds when $\lambda_3\neq 0$ and $\lambda_4=0$. On the other hand, $\lambda_3\lambda_4\neq 0$ only when
$
{\mathfrak q}=
\left(\begin{smallmatrix}
-1&\mp1&-1\\\pm1&\zeta&\pm 1\\\mp1&\pm\zeta^{2}&-1
\end{smallmatrix}\right)
$ and $\widetilde{\mathcal E}_{\bq}\neq0$ by \texttt{ufo4c.log}.

For $\Dchainthree{-1}{\zeta }{-1}{-\zeta }{-1}$, the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_3$, with defining relations
\begin{align*}
 y_1^2&=\lambda_1; & y_2^2&=\lambda_2; & [[y_{32},y_{321}]_c,y_2]_c&=\lambda_5; &
y_3^2&=\lambda_3; &  y_{13}&=\lambda_4.
\end{align*}
Here $\lambda_5\neq0$ only if ${\mathfrak q}=\left(
\begin{smallmatrix}
-1& -\texttt{c}^2\zeta & -\texttt{c}^{-3}\\ -\texttt{c}^{-2}& -1 & \texttt{c}\\ -\texttt{c}^3 & -\texttt{c}^{-1}\zeta & -1
\end{smallmatrix}
\right)$, some $\texttt{c}\in\Bbbk^\times$. Otherwise $\widetilde{\mathcal E}_{\bq}\neq 0$ by making $y_{23}=0$. When $\lambda_5\neq0$ is the only nonzero scalar then $\widetilde{\mathcal E}_{\bq}\neq0$ by \texttt{ufo4d.log}. Then $\widetilde{\mathcal E}_{\bq}\neq 0$ by Lemma \ref{lem:nonzero-k<l}.

For $\Dchainthree{-1}{\zeta }{-\zeta }{-\overline{\zeta}}{-1}$, the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_3$, with defining relations
\begin{align*}
 y_1^2&=\lambda_1; &  y_{2221}&=\lambda_2; & y_3^2&=\lambda_3; &  y_{13}&=\lambda_4 &  y_{223}&=0.
\end{align*}
Here $\lambda_2\neq 0$ only if ${\mathfrak q}=\left(
\begin{smallmatrix}
-1& -\zeta & *\\ -1& -\zeta & *\\ *&* & -1
\end{smallmatrix}
\right)$. If $\lambda_2=0$, $\widetilde{\mathcal E}_{\bq}\neq 0$, by making $y_2=0$.
A similar argument shows that $\widetilde{\mathcal E}_{\bq}\neq 0$ when $\lambda_2\neq0$: $\widetilde{\mathcal E}_{\bq}$ arises as the linking of a rank-two case (generated by $y_1,y_2$) and a rank-one case.

For $\Dchainthree{-1}{\overline{\zeta}}{\overline{\zeta}}{-\overline{\zeta}}{-1}$, the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_3$, with defining relations
\begin{align*}
 y_1^2&=\lambda_1; & y_2^3&=\lambda_2; & y_3^2&=\lambda_3; &  y_{13}&=\lambda_4; & [y_{223}, y_{23}]_c&=\lambda_5;
\end{align*}
\vspace*{-0.7cm}
\begin{align*}
[y_1,y_{223}]_c+q_{23}[y_{123},y_2]_c -q_{12} y_2y_{123}=0,
\end{align*}
with $\lambda_2\lambda_4=0$. We may project onto a case of smaller rank if $\lambda_4=\lambda_1=0$ or $\lambda_4=\lambda_3=\lambda_5=0$, so $\widetilde{\mathcal E}_{\bq}\neq 0$ in these cases. The same holds if $\lambda_2=\lambda_5=0$. Now, if  $\lambda_1\lambda_4\neq0$, then $\lambda_5=0$; hence $\widetilde{\mathcal E}_{\bq}\neq0$.

If $\lambda_2\lambda_5\neq0$ or $\lambda_2\lambda_3\neq0$ and besides $\lambda_1\neq0$, then ${\mathfrak q}=\left(
\begin{smallmatrix}
-1& \zeta^2 & \pm1\\ 1& \zeta^2 & 1\\ \pm1 & \zeta^2 & -1
\end{smallmatrix}
\right)$.
In any case $\widetilde{\mathcal E}_{\bq}\neq0$: on the one hand this holds if $\lambda_5\neq 0$ is the unique nonzero scalar, by making $y_1=0$, and on the other when $\lambda_4=\lambda_5=0$ by \cite[Lemma 5.16]{AAGMV}; hence $\widetilde{\mathcal E}_{\bq}\neq0$ using Lemma \ref{lem:nonzero-k<l}.

For $\Dchainthree{-1}{\overline{\zeta}}{\zeta }{-\zeta }{-1}$, the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_3$, with defining relations
\begin{align*}
 y_1^2&=\lambda_1; & y_2^3&=\lambda_2; & y_3^2&=\lambda_3;  \\
y_{221}&=0; & y_{13}&=\lambda_4; & [y_{223},y_{23}]_c&=\lambda_5.
\end{align*}
Here $\widetilde{\mathcal E}_{\bq}\neq 0$ by making $y_{12}=0$.

For $\Dtriangle{-1}{-1}{\zeta }{-1}{\overline{\zeta}}{-\zeta }$, the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_3$, with defining relations
\begin{align*}
 y_1^2&=\lambda_1; & y_2^3&=\lambda_2; & y_3^2&=\lambda_3; &  [y_{221},y_{21}]_c&=\lambda_4;
& y_{13}^2&=\lambda_5; &  y_{223}&=0;
\end{align*}
\vspace*{-0.7cm}
\begin{align*}
y_{(13)}+\frac{q_{23}(1-\zeta)}{2}[y_{13},y_2]_c-q_{12}(1-\overline{\zeta})y_2y_{13}=0.
\end{align*}
We can project onto a case of smaller rank and thus show that $\widetilde{\mathcal E}_{\bq}$ is nonzero whenever ${\mathfrak q}\neq\left(
\begin{smallmatrix}
-1& \zeta & \pm1\\ -1& \zeta & 1\\ \mp1 & \zeta^2 & -1
\end{smallmatrix}
\right)$. Still, $\widetilde{\mathcal E}_{\bq}\neq0$, by \texttt{ufo4h.log}.

For $\Dtriangle{-\zeta }{\zeta }{-1}{-\overline{\zeta}}{\overline{\zeta}}{-\overline{\zeta}}$, the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_3$, with defining relations
\begin{align*}
y_{113}&=\lambda_1; & y_2^2&=\lambda_2; & y_3^3&=\lambda_3 & y_{112}&=0; &  y_{332}&=0;
\end{align*}
\vspace*{-0.7cm}
\begin{align*}
y_{(13)}+q_{23}(1-\overline{\zeta})[y_{13},y_2]_c-q_{12}(1-\overline{\zeta})y_2y_{13}=0.
\end{align*}
Here $\lambda_1\neq 0$ only if ${\mathfrak q}=\left(
\begin{smallmatrix}
-\zeta & * & \zeta\\ *& -1 & *\\ -\zeta&* & \zeta
\end{smallmatrix}
\right)$ and $\lambda_1\lambda_2=0$. Hence $\widetilde{\mathcal E}_{\bq}\neq0$, by projecting onto a rank 2 case.

\subsubsection{\ Type $\mathfrak{ufo}(5)$, $\zeta \in \mathrm{G}'_3$}\label{subsec:type-ufo(5)}
The Weyl groupoid has six objects.

For $\Dchainfour{-\zeta }{-\overline{\zeta}}{-\zeta }{-\overline{\zeta}}{-\zeta }{-\overline{\zeta}}{\zeta }$, the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining relations
\begin{align*}
 y_{112}&=0; & y_{221}&=0; & y_{4}^3&=\lambda_1;\\
 y_{223}&=0; &  y_{332}&=0; & y_{334}&=\lambda_2; & y_{ij}&=0, \, i<j, \, \widetilde{q}_{ij}=1.
\end{align*}
We can project onto a case of smaller rank, hence $\widetilde{\mathcal E}_{\bq}\neq 0$.

For $\Dchainfour{-\zeta }{-\overline{\zeta}}{-\zeta }{-\overline{\zeta}}{-1}{-1}{\zeta }$, the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining relations
\begin{align*}
 y_{112}&=0; & y_{221}&=0; & [y_{443},& y_{43}]_c=0;  & y_{ij}&=0, \, i<j, \, \widetilde{q}_{ij}=1;\\
y_{223}&=0; &  y_3^2&=\lambda_1; & y_{4}^3&=\lambda_2; & [[y_{43},&y_{432}]_c,y_3]_c=0.
\end{align*}
We can project onto a case of smaller rank, hence $\widetilde{\mathcal E}_{\bq}\neq 0$.

For $\Dchainfour{-\zeta }{-\overline{\zeta}}{\zeta }{\overline{\zeta}}{-1}{-\overline{\zeta}}{-\zeta }$, the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining relations
\begin{align*}
 y_{112}&=\lambda_1; & y_{443}&=0; & y_{ij}&=0, \, i<j, \, \widetilde{q}_{ij}=1; &
y_2^3&=\lambda_2; & y_{3}^2&=\lambda_3.
\end{align*}
We can project onto a case of smaller rank, hence $\widetilde{\mathcal E}_{\bq}\neq 0$.

For $\Dchainfour{-\zeta }{-\overline{\zeta}}{-\zeta }{-\overline{\zeta}}{-1}{-\overline{\zeta}}{-\zeta }$, the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining relations
\begin{align*}
 y_{112}&=y_{221}=y_{223}=y_{443}=0; & y_{ij}&=0, \, i<j, \, \widetilde{q}_{ij}=1; &
y_{3}^2&=\lambda_1.
\end{align*}
We can project onto a case of smaller rank, hence $\widetilde{\mathcal E}_{\bq}\neq 0$.

For $\Drightofway{-\zeta }{-\overline{\zeta}}{-1}{-\zeta }{\overline{\zeta}}{-1}{-1}{\zeta }$, the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining relations
\begin{align*}
[y_{334},&y_{34}]_c=\lambda_1; & y_{112}&=0; & y_{2}^2&=\lambda_2; & y_{ij}&=0, \, i<j, \, \widetilde{q}_{ij}=1;\\
[y_{124},& y_2]_c=0; & y_{332}&=0; & y_3^3&=\lambda_3; & y_{4}^2&=\lambda_4;
\end{align*}
\vspace*{-0.7cm}
\begin{align*}
y_{(24)}-2q_{34}\zeta[y_{24},y_3]_c-2q_{23}y_3y_{24}=0.
\end{align*}
We can project onto a case of smaller rank, hence $\widetilde{\mathcal E}_{\bq}\neq 0$.

For $\Drightofway{-\zeta }{-\overline{\zeta}}{-1}{-\zeta }{\zeta }{-1}{-\zeta }{-1}$, the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining relations
\begin{align*}
y_{2}^2&=\lambda_1; & y_{3}^2&=\lambda_2;  & y_4^2&=\lambda_3; & y_{ij}&=0, \, i<j, \, \widetilde{q}_{ij}=1;
\end{align*}
\vspace*{-0.7cm}
\begin{align*}
y_{112}&=0; &  [y_{124},&y_2]_c=0; & y_{(24)}+q_{34}\overline{\zeta} [y_{24},y_3]_c+q_{23}\overline{\zeta} y_3y_{24}=0.
\end{align*}
We can project onto a case of smaller rank, hence $\widetilde{\mathcal E}_{\bq}\neq 0$.

\subsubsection{\ Type $\mathfrak{ufo}(6)$, $\zeta \in \mathrm{G}'_4$}\label{subsec:type-ufo(6)}
The Weyl groupoid has eight objects.

For $\Dchainfour{\overline{\zeta} }{\zeta }{-1}{\overline{\zeta} }{\zeta }{\zeta }{\overline{\zeta} }$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining relations
\begin{align*}
 y_{112}&=\lambda_1;  & y_{2}^2&=\lambda_2;  &   y_3^4&=\lambda_7 & y_{332}&=\lambda_3;   \\
  y_{13}&=\lambda_5 &   y_{443}&=0; & y_{14}&=y_{24}=0;  & [y_{(13)},& y_2]_c=\lambda_4;
\end{align*}
\vspace*{-0.7cm}
 \begin{align*}
    [[[y_{(24)}, y_3]_c,y_3]_c,y_3]_c=\lambda_6.
\end{align*}
Here $\lambda_1\lambda_3=\lambda_1\lambda_4=\lambda_1\lambda_5=0=\lambda_1\lambda_6$. Hence we can project onto a case of smaller rank and thus $\widetilde{\mathcal E}_{\bq}\neq 0$. See \texttt{ufo6a.log} for the deformation of the last relation.

For $\Dchainfour{-1}{\overline{\zeta} }{-1}{\zeta }{-1}{\zeta }{\overline{\zeta} }$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining relations
\begin{align*}
y_1^2&=\lambda_1; &   y_2^2&=\lambda_2; & y_3^2&=\lambda_3; & [y_{(13)},&y_2]_c=\lambda_4; \\
y_{443}&=\lambda_5; & y_{13}&=\lambda_6 & y_{14}&=y_{24}=0; & [[y_{23},&[y_{23},y_{(24)}]_c]_c,y_3]_c=0.
 \end{align*}
If $\lambda_5=0$, then $\widetilde{\mathcal E}_{\bq}\neq0$ by making $y_4=0$. Similarly if $\lambda_5$ is the unique nonzero scalar. The general case now follows from Lemma \ref{lem:nonzero-k<l}.

For $\Dchainfour{-1}{\zeta }{\overline{\zeta} }{\zeta }{-1}{\zeta }{\overline{\zeta} }$
the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining relations
\begin{align*}
y_1^2&=\lambda_1;  & y_{221}&=\lambda_2; & y_3^2&=\lambda_3; & y_{223}&=\lambda_4; \\
 y_{443}&=\lambda_5; & y_{13}&=\lambda_6;  &  y_{14}&=y_{24}=0.
 \end{align*}
If $\lambda_5\lambda_4\neq 0$, then $\lambda_1=\lambda_2=\lambda_6=0$ and hence we can project by making $y_1=0$, so $\widetilde{\mathcal E}_{\bq}\neq 0$. If $\lambda_5\neq 0$ and $\lambda_4=0$, then $\widetilde{\mathcal E}_{\bq}\neq 0$ by making $y_{23}=0$. If $\lambda_5=0$, then $\widetilde{\mathcal E}_{\bq}\neq 0$ by making $y_4=0$.

For $\Dchainfour{-1}{\overline{\zeta} }{\zeta }{-1}{-1}{\zeta }{\overline{\zeta} }$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining relations
\begin{align*}
 y_1^2&=\lambda_1; & y_{221}&=\lambda_2 & y_3^2&=\lambda_3; & y_{13}&=\lambda_4; \\
 y_{443}&=\lambda_5; &    y_{2223}&=0; & y_{14}&=0 & y_{24}&=0;
 \end{align*}
 \vspace*{-0.7cm}
\begin{align*}
[y_2, & [y_{(24)},y_3]_c]_c = \frac{q_{23}q_{43}}{1+\zeta}[y_{23},y_{(24)}]_c+(\zeta-1)q_{23}q_{24} y_{(24)}y_{23}.
\end{align*}
If $\lambda_5=0$, then $\widetilde{\mathcal E}_{\bq}\neq0$ by making $y_4=0$. Similarly if $\lambda_5$ is the unique nonzero scalar. The general case now follows from Lemma \ref{lem:nonzero-k<l}.

For $\Dchainfour{-1}{\zeta }{-1}{-1}{-1}{\zeta }{\overline{\zeta} }$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining relations
\begin{align*}
y_1^2&=\lambda_1; & y_2^2&=\lambda_2;  &  y_3^2&=\lambda_3; &  y_{443}&=\lambda_4; \\
y_{23}^2&=\lambda_5; & y_{13}&=\lambda_6 & y_{14}&=y_{24}=0; &  [[y_{12},&y_{(13)}]_c, y_2]_c=\lambda_7.
\end{align*}
If $\lambda_4=0$, then $\widetilde{\mathcal E}_{\bq}\neq0$ by making $y_4=0$. Similarly if $\lambda_4$ is the unique nonzero scalar. The general case now follows from Lemma \ref{lem:nonzero-k<l}.

For $\Drightofway{\overline{\zeta} }{\zeta }{-1}{\overline{\zeta} }{-1}{\zeta }{\overline{\zeta} }{-1}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining relations
\begin{align*}
 y_{112}&=\lambda_1; & y_2^2&=\lambda_2; & y_3^2&=\lambda_3; & y_{443}&=\lambda_4; & y_{442}&=\lambda_5; \\
y_{23}^2&=\lambda_6;   &  y_{14}&=\lambda_7; & [y_{124},&y_2]_c=\lambda_8; & y_{13}&=0;
\end{align*}
\vspace*{-0.7cm}
\begin{align*}
y_{(24)}-q_{34}\zeta [y_{24},y_3]_c-q_{23}(1+\zeta)y_3y_{24}=0.
\end{align*}
We analyze the cases in which we cannot project onto a case of smaller rank, notice that, for instance, $\lambda_1\lambda_5\neq 0$ or $\lambda_1\lambda_5\neq 0$, imply that $\lambda_3=\lambda_4=\lambda_6=0$ and we can make $y_3=0$. Similarly if $\lambda_7\neq 0$ or $\lambda_8\neq 0$.

Finally, if $\lambda_1\lambda_4\neq 0$, then
${\mathfrak q}=\left(\begin{smallmatrix}
\overline{\zeta} & -1 & 1 & \pm1\\
\overline{\zeta} & -1 & -1 & \overline{\zeta} \\
1 & 1 & -1 & \zeta \\
\pm1  & 1 & -1 & \zeta \\
\end{smallmatrix}\right)$ and $\lambda_5=\lambda_7=\lambda_8=0$; we could also have $\lambda_2\lambda_3\lambda_6\neq 0$. In any case, $\widetilde{\mathcal E}_{\bq}\neq 0$ by \texttt{ufo6f.g}.

For $\Drightofway{-1}{\overline{\zeta} }{\zeta }{\overline{\zeta} }{-1}{-1}{\overline{\zeta} }{-1}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining relations
\begin{align*}
 y_1^2&=\lambda_1; & y_{13}&=\lambda_2 & y_3^2&=\lambda_3; & y_4^2&=\lambda_4; \\
 y_{221}&=\lambda_5; & y_{224}&=\lambda_6 & y_{14}&=\lambda_7; & y_{2223}&=0;
\end{align*}
\vspace*{-0.7cm}
\begin{align*}
y_{(24)}-q_{34}\zeta [y_{24},y_3]_c-q_{23}(1+\zeta)y_3y_{24}=0.
\end{align*}
Here $\widetilde{\mathcal E}_{\bq}\neq0$ as we can quotient by $y_{23}$, $y_{34}$.

For $\Drightofway{-1}{\zeta }{-1}{\overline{\zeta} }{-1}{-1}{\overline{\zeta} }{-1}$ the algebra $\widetilde{\mathcal E}_{\bq}$ is generated by $y_i$, $i\in\I_4$, with defining relations
\begin{align*}
 y_1^2&=\lambda_1; & y_2^2&=\lambda_2; & y_3^2&=\lambda_3; &  y_4^2&=\lambda_4; & [y_{124},& y_2]_c=\lambda_5;  \\
& &  y_{14}&=\lambda_6; & y_{23}^2&=\lambda_7;  &  y_{13}&=\lambda_8;  & [[y_{12},&y_{(13)}]_c,y_2]_c=\lambda_9;
\end{align*}
\vspace*{-0.7cm}
\begin{align*}
y_{(24)}-q_{34}\zeta [y_{24},y_3]_c-q_{23}(1+\zeta)y_3y_{24}=0.
\end{align*}
If $\lambda_5=0$, then $\widetilde{\mathcal E}_{\bq}\neq 0$ since we can quotient by $y_{24}$, $y_{34}$. Similarly if $\lambda_4$ is the unique nonzero scalar. The general case now follows from Lemma \ref{lem:nonzero-k<l}.

\subsubsection{Type $\mathfrak{ufo}(7)$, $\zeta \in \mathrm{G}'_{12}$}\label{subsec:type-ufo(7)}
The Weyl groupoid has five objects but two of them are obtained with $\zeta^5$
instead of $\zeta$. Hence we focus in the remaining three cases.

For $\Dchaintwo{-\overline{\zeta}^{\,2}}{-\zeta ^3}{-\zeta ^2}$,
$\widetilde{\mathcal E}_{\bq}$ is generated by $y_1, y_2$ with defining relations
\begin{align*}
y_1^3&=\lambda_1; & y_2^3&=\lambda_2; &  [y_1,y_{122}]_c -\frac{\zeta^{10}(1+\zeta)q_{12}}{1+\zeta^3}y_{12}^2&=0,
\end{align*}
with $\lambda_1\lambda_2=0$. This is nonzero by \cite[Lemma 5.16]{AAGMV}.

For $\Dchaintwo{-\overline{\zeta}^{\,2}}{\overline{\zeta}}{-1}$,
$\widetilde{\mathcal E}_{\bq}$ is generated by $y_1, y_2$ with defining relations
\begin{align*}
y_1^3&=\lambda_1; & y_2^2&=\lambda_2; & [[y_{112},y_{12}]_c,y_{12}]_c&=0,
\end{align*}
with $\lambda_1\lambda_2=0$. The last relation is undeformed in any case, see
\texttt{ufo(7).log}. This algebra is nonzero by \cite[Lemma 5.16]{AAGMV}.

For $\Dchaintwo{-\zeta ^3}{\zeta}{-1}$, $\widetilde{\mathcal E}_{\bq}$ is generated by $y_1, y_2$ with defining relations
\begin{align*}
y_1^4&=\lambda_1; & y_2^2&=\lambda_2; &  [ y_{112},y_{12}]_c&=0,
\end{align*}
with $\lambda_1\lambda_2=0$. This is nonzero by \cite[Lemma 5.16]{AAGMV}.

\subsubsection{Type $\mathfrak{ufo}(8)$, $\zeta \in \mathrm{G}'_{12}$}\label{subsec:type-ufo(8)}
The Weyl groupoid has three objects.

For $\Dchaintwo{-\zeta ^2}{\zeta }{-\zeta ^2}$, $\widetilde{\mathcal E}_{\bq}$ is generated by $y_1, y_2$ with defining relations
\begin{align*}
y_1^3&=\lambda_1; & y_2^3&=\lambda_2; & [y_1,y_{122}]_c-(1+\zeta+\zeta^2)\zeta^4q_{12}y_{12}^2&=0,
\end{align*}
with $\lambda_1\lambda_2=0$. This algebra is nonzero by \cite[Lemma 5.16]{AAGMV}.

For $\Dchaintwo{-\zeta ^2}{\zeta ^3}{-1}$, $\widetilde{\mathcal E}_{\bq}$ is generated by $y_1, y_2$ with defining relations
\begin{align*}
y_1^3&=\lambda_1; & y_2^2&=\lambda_2; & [[y_{112},y_{12}]_c,y_{12}]_c&=0.
\end{align*}
with $\lambda_1\lambda_2=0$. The last relation is undeformed in any case, see
\texttt{ufo8.log}. This algebra is nonzero by \cite[Lemma 5.16]{AAGMV}.

For $\Dchaintwo{-\overline{\zeta}}{-\zeta ^3}{-1}$, $\widetilde{\mathcal E}_{\bq}$ is generated by $y_1, y_2$ with defining relations
\begin{align*}
y_{11112}&=0; & y_2^2&=\lambda_2; &   [y_{112},y_{12}]_c&=0.
\end{align*}
This algebra is nonzero by \cite[Lemma 5.16]{AAGMV}.

\subsubsection{Type $\mathfrak{ufo}(9)$, $\zeta \in \mathrm{G}'_{24}$}\label{subsec:type-ufo(9)}
The Weyl groupoid has four objects.

For $\Dchaintwo{\zeta ^6}{-\overline{\zeta}}{\ \ -\overline{\zeta}^{\, 4}}$, $\widetilde{\mathcal E}_{\bq}$ is generated by $y_1, y_2$ with defining relations
\begin{align*}
y_1^4&=\lambda_1; & y_2^3&=\lambda_2; & [y_1,y_{122}]_c - \frac{1+\zeta^7}{1+\zeta}\zeta^{10}q_{12}y_{12}^2&=0,
\end{align*}
with $\lambda_1\lambda_2=0$. This algebra is nonzero by \cite[Lemma 5.16]{AAGMV}.

For $\Dchaintwo{\zeta ^6}{\zeta }{\overline{\zeta}}$, $\widetilde{\mathcal E}_{\bq}$ is generated by $y_1, y_2$ with defining relations
\begin{align*}
y_1^4&=\lambda_1; \qquad  y_{221}=0; \\
[[y_{112}&,y_{12}]_c,y_{12}]_c= -\lambda_1q_{21}^{-1}(2\zeta+\zeta^8+2\zeta^{17}-\zeta^{22}) y_2^3 .
\end{align*}
See \texttt{ufo9b.log} and \texttt{ufo9b2.log}. This algebra is nonzero, as it projects onto $\Bbbk\langle y_1|y_1^4=\lambda_1 \rangle$.

For $\Dchaintwo{-\overline{\zeta}^{\, 4}}{\zeta ^5}{-1}$, $\widetilde{\mathcal E}_{\bq}$ is generated by $y_1, y_2$ with defining relations
\begin{align*}
y_1^3&=\lambda_1; & y_2^2&=\lambda_2;  & [y_{112},[[y_{112},y_{12}]_c,y_{12}]_c]_c -\alpha [y_{112},y_{12}]_c^2 &=0,
\end{align*}
for $\alpha=\frac{1+\zeta+\zeta^6+2\zeta^7+\zeta^{17}}{1+\zeta^4+\zeta^6+\zeta^{11}} \zeta^{9}q_{12}$. Here, $\lambda_1\lambda_2=0$.
The last relation remains undeformed, see
\texttt{ufo9c.log}. This algebra is nonzero by \cite[Lemma 5.16]{AAGMV}.

For $\Dchaintwo{\zeta }{\overline{\zeta}^{\, 5}}{-1}$, $\widetilde{\mathcal E}_{\bq}$ is generated by $y_1, y_2$ with defining relations
\begin{align*}
y_2^2&=\lambda_1; & y_{1111112}&=0; & [y_{112},y_{12}]_c&=0
\end{align*}
This algebra is nonzero by \cite[Lemma 5.16]{AAGMV}.
The last relation remains undeformed by direct computation.

\subsubsection{Type $\mathfrak{ufo}(10)$, $\zeta\in \mathrm{G}'_{20}$}\label{subsec:type-ufo(10)}
The Weyl groupoid has four objects but two of them are obtained with $-\zeta$
instead of $\zeta$. Hence we focus in just two cases.

For $\Dchaintwo{\zeta }{\overline{\zeta}^{\, 3}}{-1}$, $\widetilde{\mathcal E}_{\bq}$ is generated by $y_1, y_2$ with defining relations
\begin{align*}
y_2^2&=\lambda_1;  &  y_{11112}&=0;  & [[[y_{112},y_{12}]_c,y_{12}]_c,y_{12}]_c&=0.
\end{align*}
This algebra is nonzero by \cite[Lemma 5.16]{AAGMV}.  The last relation remains undeformed by
\texttt{ufo10a.log}.

For $\Dchaintwo{-\overline{\zeta}^{\, 2}}{\zeta ^3}{-1}$, $\widetilde{\mathcal E}_{\bq}$ is generated by $y_1, y_2$ with defining relations
\begin{align*}
y_1^5&=\lambda_1; & y_2^2&=\lambda_2 &
[y_1,[y_{112},y_{12}]_c]_c+\frac{1-\zeta^{17}}{1-\zeta^2}q_{12}y_{112}^2&=0,
\end{align*}
with $\lambda_1\lambda_2=0$. The last relation remains undeformed, see  \texttt{ufo10c.log}, and thus this algebra is nonzero by \cite[Lemma 5.16]{AAGMV}.

\subsubsection{Type $\mathfrak{ufo}(11)$, $\zeta\in \mathrm{G}'_{15}$}\label{subsec:type-ufo(11)}
The Weyl groupoid has four objects.

For $\Dchaintwo{-\zeta }{-\overline{\zeta}^{\, 3}}{\zeta ^5}$, $\widetilde{\mathcal E}_{\bq}$ is generated by $y_1, y_2$ with defining relations
\begin{align*}
y_2^3&=\lambda_2; & y_{11112}&=0; & [[y_{112},y_{12}]_c,y_{12}]_c &=0;
&
[y_1,y_{122}]_c +\alpha y_{12}^2&=0,
\end{align*}
for $\alpha=\frac{1+\zeta^{13}}{1+\zeta^{12}} \zeta^{10}q_{12}$. This algebra is nonzero by \cite[Lemma 5.16]{AAGMV}.
 The third relation  is undeformed by \texttt{ufo11a1.log}.
  We recall that the fourth one is primitive in $T(V)$.

For $\Dchaintwo{\zeta ^3}{-\zeta ^4}{-\overline{\zeta}^{\,4}}$, $\widetilde{\mathcal E}_{\bq}$ is generated by $y_1, y_2$ with defining relations
\begin{align*}
[y_1,[y_{112},y_{12}]_c]_c &=\frac{1-\zeta^2}{1+\zeta^7}\zeta^9q_{12} y_{112}^2; \qquad y_1^5=\lambda_1; \qquad  y_{221}=0; \\
[[[y_{112},y_{12}]_c,y_{12}]_c,y_{12}]_c&= \lambda_1q_{21}(\zeta+\zeta^4+3\zeta^8+4\zeta^{13}-3\zeta^{14})y_2^4,
\end{align*}
See \texttt{ufo11b.log} for the rule of deformation of the bottom quantum Serre relation.
This algebra is nonzero as it projects onto $\Bbbk\langle y_1| y_1^5=\lambda_1 \rangle$.

For $\Dchaintwo{\zeta ^5}{-\overline{\zeta}^{\, 2}}{-1}$, $\widetilde{\mathcal E}_{\bq}$ is generated by $y_1, y_2$ with defining relations
\begin{align*}
 y_1^3&=\lambda_1; & y_2^2&=\lambda_2; & [[[y_{112},y_{12}]_c,y_{12}]_c,y_{12}]_c& =0,
\end{align*}
with $\lambda_1\lambda_2=0$. The third relation is undeformed by \texttt{ufo11c.log} and thus
 this algebra is nonzero by  \cite[Lemma 5.16]{AAGMV}.

For $\Dchaintwo{\zeta ^3}{-\zeta ^2}{-1}$, $\widetilde{\mathcal E}_{\bq}$ is generated by $y_1, y_2$ with defining relations
\begin{align*}
 y_1^5&=\lambda_1; & y_2^2&=\lambda_2; &  [y_{112},y_{12}]_c&=0; &  [[y_{1112},y_{112}]_c,y_{112}]_c&=0,
\end{align*}
$\lambda_1\lambda_2=0$. The third relation is undeformed by direct computation.
As for the fourth, see  \texttt{ufo11d.log}.
Then this algebra is nonzero \cite[Lemma 5.16]{AAGMV}.

\subsubsection{Type $\mathfrak{ufo}(12)$, $\zeta\in \mathrm{G}'_{7}$}\label{subsec:type-ufo(12)}
The Weyl groupoid has two objects.

For $\Dchaintwo{-\zeta }{-\overline{\zeta}^{\, 3}}{-1}$, $\widetilde{\mathcal E}_{\bq}$ is generated by $y_1, y_2$ with defining relations
\begin{align*}
 y_{11112}&=0; & y_2^2&=\lambda_2; &[y_{112},[[y_{112},y_{12}]_c,y_{12}]_c]_c
 -\alpha[y_{112},y_{12}]_c^2& =0,
\end{align*}
for $\alpha=-q_{12}\frac{-\zeta+3\zeta^2+3\zeta^3-\zeta^4+3\zeta^6}{-2\zeta+2\zeta^3-\zeta^5+\zeta^6}$.
By \texttt{ufo12a.log}, the last relation is undeformed.
The algebra is nonzero as it projects onto $\Bbbk\langle y_2^2|y_2^2=\lambda_2\rangle$.

For $\Dchaintwo{-\overline{\zeta}^{\, 2}}{-\zeta ^3}{-1}$, $\widetilde{\mathcal E}_{\bq}$ is generated by $y_1, y_2$ with defining relations
\begin{align*}
 y_{1111112}&=0; & y_2^2&=\lambda_2; &
[y_1,[y_{112},y_{12}]_c]_c +q_{12}\frac{1+\zeta^4}{1-\zeta^2}y_{112}^2 &=0.
\end{align*}
The last relation is undeformed by direct computation. The algebra is nonzero as it projects onto $\Bbbk\langle y_2^2|y_2^2=\lambda_2\rangle$.

\subsection{The algebras $\widetilde{\mE}_{\bq}$}

\subsubsection{} To define $\widetilde{\mathcal{E}}_{{\mathfrak q}}(\boldsymbol\lambda)$, we follow \eqref{eqn:E(lambda)}, by deforming the corresponding defining relations of $\mathfrak{B}_{\mathfrak{q}}$, as computed in \cite{A-nichols} for each diagram in the classification of \cite{H-classif}. This is a case-by-case analysis of all Nichols algebras following the explicit list given in \cite{AA}. Namely, given a presentation of a given Nichols algebra $\mathfrak{B}(V)$ in \cite{AA} by a set of relations ${\mathcal G}$, we keep only those relations which are not powers of Cartan root vectors (which is, in a generic way, a presentation of the distinguished pre-Nichols algebra $\widetilde{\mathfrak{B}}(V)$ from \cite{A-distinguished}), we stratify this set $\widetilde{{\mathcal G}}$ and inductively deform each relation according to \eqref{eqn:E(lambda)}.

\subsection{Tools and techniques}
For each ${\mathfrak q}$ we show that $\widetilde{\mathcal E}_{\bq} \neq0$ using a combination of the following results and resources.

\subsubsection{}\label{subsubsec:gap} For diagrams of small rank we explicitly compute the Gr\"obner basis of the defining ideal of $\widetilde{\mathcal E}_{\bq}$. This is achieved using the computer program \cite{GAP}, together with the package \texttt{GBNP} by \cite{CG}. The \texttt{log} files are stored in the authors' webpages:
\texttt{http://www.famaf.unc.edu.ar/$\sim$(angiono|aigarcia)}.

\subsubsection{}\label{subsubsec:oldresults} We invoke some results from previous articles. The case in which only the powers of simple roots are deformed follows from \cite[Lemma 5.16]{AAGMV}. On the other hand, the case with the nontrivial deformations of the quantum Serre relations were investigated in \cite[Proposition 3.2]{AGI}.

\subsubsection{}\label{subsubsec:cut} Assume that all the generalized quantum Serre relations involving two letters $y_i$, $y_j$, $i<j\in \I$ and $q_{ij}q_{ji}\neq 1$ (i.e. adjacent vertices), are not deformed. Let ${\mathcal G}[ij]\subset {\mathcal G}$ be the subset containing these relations.

First, assume further that there is no $k\neq i,j\in \I$ adjacent to both $i$ and $j$. Let ${\mathfrak q}'=(q_{st}')_{s,t\in\I}$ be the matrix such that $q_{st}'=q_{st}$ when $(s,t)\neq(j,i)$ and $q_{ji}'=q_{ij}^{-1}$. Hence ${\mathfrak q}'$ has two connected components arising by cutting the original diagram between $i$ and $j$. Thus the set of defining relations of $\mathfrak{B}_{{\mathfrak q}'}$ is ${\mathcal G}'=\left({\mathcal G}\setminus {\mathcal G}[ij]\right)\cup \{x_{ij}\}$. Now $\widetilde{\mathcal E}_{\bq}(\boldsymbol\lambda)/\langle y_{ij}\rangle$ projects onto $\widetilde{\mathcal{E}}_{{\mathfrak q}'}(\boldsymbol\lambda')$ for $\boldsymbol\lambda'=(\lambda'_r)_{r\in {\mathcal G}'}$ with $\lambda'_r=\lambda_r $ if $r\in{\mathcal G} \setminus{\mathcal G}[ij]$ and $\lambda'_{ij}=0$.  In this case we show that $\widetilde{\mathcal E}_{\bq}(\boldsymbol\lambda)\neq 0$ using a recursive argument.

Second, assume $i<j<k\in \I$ are pairwise adjacent and all the generalized quantum Serre relations involving either the letters $y_i$ and $y_j$ or $y_i$ and $y_k$ are not deformed. In this case, we proceed in a similar way by cutting the diagram between $i$ and $j$, and also between $i$ and $k$.

\subsubsection{}\label{subsubsec:projection} Assume that all the relations involving a letter $y_i$, $i\in \I$, are not deformed. Let ${\mathcal G}[i]\subset {\mathcal G}$ be the subset containing these relations. Let ${\mathfrak q}'$ be the submatrix of ${\mathfrak q}$ obtained by removing the $i$-th row and column. Now the set of relations of $\mathfrak{B}_{{\mathfrak q}'}$ is ${\mathcal G}'={\mathcal G}\setminus {\mathcal G}[i]$ and $\widetilde{\mathcal E}_{\bq}(\boldsymbol\lambda)/\langle y_{i}\rangle$ projects onto $\widetilde{\mathcal{E}}_{{\mathfrak q}'}(\boldsymbol\lambda')$,  $\boldsymbol\lambda'=(\lambda_r)_{r\in{\mathcal G}'}$. Again, a recursive argument shows that $\widetilde{\mathcal E}_{\bq}(\boldsymbol\lambda)\neq 0$.

\subsubsection{}\label{subsubsec:casebycase} In many cases, we can restrict to the case in which a single relation $r\in{\mathcal G}$ is deformed. Then an iterative application of Lemma \ref{lem:nonzero-k<l} shows that $\widetilde{\mathcal E}_{\bq}\neq 0$ by deforming one relation at a time. In each case, we can restrict to the subdiagram involving just the letters in the relation, using the technique described in \S \ref{subsubsec:projection}.

\section{Introduction}\label{sec:intro}

The program for the classification of finite-dimensional pointed Hopf algebra with abelian coradical $\Bbbk\Gamma$ was initiated by N.~Andruskiewitsch and H.J.~Schneider in \cite{AS-lift-meth}, where they proposed a {\it lifting method} to deal with this problem. The method proved to be very fruitful as they completed the classification in the case when $|\Gamma|$ was not divisible by small primes in \cite{AS-annals}.

In \cite{AAG}, we proposed an strategy to compute all liftings of braided vector spaces of diagonal type $V\in\mathfrak{YD-}$, that is all Hopf algebras $\mathfrak{u}$ such that $\gr \mathfrak{u}=\mathfrak{B}(V)\# H$. This is a large class of Hopf algebras, which includes f.d.~pointed Hopf algebras with abelian coradical. In this article we show that this algorithm is exhaustive. Our method recovers the examples in \cite{AS-annals} and completes the classification, as it becomes {\it independent} of $\Gamma$ and holds more generally for any coradical $H$. Moreover, it also applies for Nichols algebras of infinite dimension which are finitely presented.

A remarkable feature of this method is that all liftings are constructed as cocycle deformations of $\mathfrak{u}_0=\mathfrak{B}(V)\# H$.
This has important implications on the study of tensor categories and more general Hopf algebras. On the one hand, it is equivalent to the fact that the categories of comodules of $\mathfrak{u}$ and $\mathfrak{u}_0$ are tensor equivalent; in the finite-dimensional case this equivalence extends to the categories of Yetter-Drinfeld modules ${}^{\mathfrak{u}}_{\mathfrak{u}}\mathcal{YD}$ and ${}^{\mathfrak{u}_0}_{\mathfrak{u}_0}\mathcal{YD}$. This permits the construction of new examples of finite-dimensional Nichols algebras over the basic Hopf algebra $\mathfrak{u}^\ast$ \cite{AA-new}, and \emph{a fortiori} of new Hopf algebras by bosonization with $\mathfrak{u}^\ast$.

We fix
\begin{itemize} [leftmargin=*]\renewcommand{\labelitemi}{$\circ$}
\item A cosemisimple Hopf algebra $H$.
\item A f.d.~braided vector space  of diagonal type $(V,c)$ with a principal realization in $\mathfrak{YD-}$ and such that $\dim \mathfrak{B}(V)<\infty$.
\end{itemize}
Let ${\mathcal G}\subset T(V)$ be a (minimal) set of relations defining $\mathfrak{B}(V)$. We produce, for each family $\boldsymbol\lambda=(\lambda_r)_{r\in{\mathcal G}}$ of scalars in a subset $\mathcal{R}\subset\Bbbk^{{\mathcal G}}$,
\begin{itemize} [leftmargin=*]\renewcommand{\labelitemi}{$\bullet$}
\item a Hopf algebra $\mathfrak{u}(\boldsymbol\lambda)$ such that
  \[
  \gr\mathfrak{u}(\boldsymbol\lambda)\simeq \mathfrak{B}(V)\#H.
  \]
  that is, a {\it lifting} of $V$. Moreover, $\mathfrak{u}(\boldsymbol\lambda)$ is a cocycle deformation of $\mathfrak{B}(V)\#H$.
\end{itemize}

Our main result is the following:
\begin{theorem}\label{thm:cocycle-defs}
Let $H$ be a cosemisimple Hopf algebra and let $(V,c)$ be a braided vector space  of diagonal type such that $\dim\mathfrak{B}(V)<\infty$.
Let $L$ be a Hopf algebra such that its infinitesimal braiding is given by a principal realization
$V\in \mathfrak{YD-}$. Then
\begin{enumerate}
\item There is $\boldsymbol\lambda\in\mathcal{R}$ such that $L\simeq \mathfrak{u}(\boldsymbol\lambda)$.
\item $L$ is a cocycle deformation of $\mathfrak{B}(V)\#H$.
\end{enumerate}
\end{theorem}
See \S\ref{sec:def-u} below for the definition of the algebras $\mathfrak{u}(\boldsymbol\lambda)$ and \S\ref{sec:proof-u} for comments on the proof of this result. See also Theorem \ref{thm:isos} for the description of the isomorphism classes.
\begin{proof}
Any such $L$ is a lifting of $V$ by \cite[Theorem 2]{A-nichols}. The result follows by Theorem \ref{teo:lift}.
\end{proof}

When $H$ is the group algebra of a finite abelian group $\Gamma$, Theorem \ref{thm:cocycle-defs} is equivalent to the following.
\begin{cor}\label{cor:pointed}
Let $A$ be a finite-dimensional pointed Hopf algebra with abelian coradical. Then $A$ is a cocycle deformation of $\gr A$ and there is $\boldsymbol\lambda\in\mathcal{R}$ such that $A\simeq \mathfrak{u}(\boldsymbol\lambda)$.\qed
\end{cor}

\subsection{Definition of $\mathfrak{u}(\boldsymbol\lambda)$}\label{sec:def-u}

Set $\theta=\dim V$. In this setting there exist
 \begin{align*}
g_i&\in G(H), & \chi_i&\in \Alg(H,\Bbbk), \qquad 1\leq i\leq\theta,
 \end{align*}
 such that $\Gamma \coloneqq \langle g_1,\dots,g_\theta\rangle < Z(G(H))$ and there is a linear basis $\{x_1,\dots,x_\theta\}$ of $V$ so that $x_i\in V_{g_i} ^{\chi_i}=V_{g_i}\cap V^{\chi_i}$, $1\leq i\leq \theta$, cf.~\eqref{eqn:components}. Moreover, we may assume that ${\mathcal G}$ is composed of $\mathbb{Z}^\theta$-homogeneous elements, so that if $r\in{\mathcal G}$ is of degree $(a_1,\dots,a_\theta)\in \mathbb{Z}^\theta$, then $r\in T(V)_{g_r}^{\chi_r}$, where
 \begin{align}\label{eqn:gr}
g_r&:=g_1^{a_1}\dots g_\theta^{a_\theta}\in\Gamma, & \chi_r&:=\chi_1^{a_1}\dots \chi_\theta^{a_\theta}\in \Alg(H,\Bbbk).
 \end{align}
For each $\boldsymbol\lambda=(\lambda_r)_{r\in{\mathcal G}}$ in the set
 \begin{align}\label{eqn:R}
 \mathcal{R}\coloneqq\{\boldsymbol\lambda=(\lambda_r)_{r\in {\mathcal G}}\in\Bbbk^{{\mathcal G}}:\lambda_r=0\text{ if } \chi_r\neq \epsilon\text{ or } g_r=1 \},
 \end{align} we define a Hopf algebra $\mathfrak{u}(\boldsymbol\lambda)$
 recursively as follows:
\begin{itemize} [leftmargin=*]
\item We fix an stratification ${\mathcal G}={\mathcal G}_0\sqcup\dots \sqcup G_\ell$ of the set of relations of $\mathfrak{B}(V)$ in such a way that ${\mathcal G}_{k+1}$ is a set of primitive elements of
\[\mathfrak{B}_{k+1}\coloneqq T(V)/\langle {\mathcal G}_0\cup\dots\cup{\mathcal G}_k\rangle.\]
\item We set $\mathcal{E}_0(\boldsymbol\lambda)=T(V)$ and proceed stepwise by defining an algebra
\[\mathcal{E}_{k+1}(\boldsymbol\lambda)\coloneqq \mathcal{E}_k(\boldsymbol\lambda)/\langle \gamma_k(r)-\lambda_r|r\in{\mathcal G}_{k+1}\rangle.\]
In this setting, $\mathcal{A}_k(\boldsymbol\lambda)=\mathcal{E}_k(\boldsymbol\lambda)\# H$ is a right $\mathfrak{B}_{k}\#H$-cleft object and $\gamma_k:\mathfrak{B}_{k}\#H\to \mathcal{A}_k(\boldsymbol\lambda)$ is a section.
\item We set $\mathfrak{u}_0(\boldsymbol\lambda)=T(V)$ and proceed stepwise by defining an algebra
\[\mathfrak{u}_{k+1}(\boldsymbol\lambda)\coloneqq \mathfrak{u}_k(\boldsymbol\lambda)/\langle \tilde{r}-\lambda_r(1-g_r)|r\in{\mathcal G}_{k+1}\rangle\]
in such a way that $\mathcal{A}_k(\boldsymbol\lambda)$ becomes a left $\mathfrak{u}_{k}(\boldsymbol\lambda)$-cleft object and $\tilde{r}\in\mathfrak{u}_k(\boldsymbol\lambda)$ is defined so that the coaction satisfies  $\gamma_k(r)_{(-1)}\otimes \gamma_k(r)_{(0)}=\tilde{r}\otimes 1+g_r\otimes \gamma_k(r)$. Thus,
\begin{align}\label{eqn:u}
\mathfrak{u}(\boldsymbol\lambda)\coloneqq T(V)\# H/\langle \tilde{r}-\lambda_r(1-g_r)|r\in{\mathcal G}_{k}\rangle_{0\leq k\leq\ell}
\end{align}
\end{itemize}

The explicit presentation of the algebras $\mathfrak{u}(\boldsymbol\lambda)$ is a hard computational problem. When the braiding is of Cartan type $A$, a complete description was achieved in \cite{AAG} and the Cartan type $G_2$ was completed in \cite{JG}.

\subsection{On the proof}\label{sec:proof-u}
The proof of our main theorem relies on the fact that we are able to show that, for each braiding matrix $\mathfrak{q}$, a given algebra $\widetilde{\mE}_{\bq}(\boldsymbol\lambda)$ is nonzero. This is a technical condition pointed in our previous work \cite{AAG} as an obstruction to prove that the list of algebras $\mathfrak{u}(\boldsymbol\lambda)$ was exhaustive.

This is achieved in Theorem \ref{thm:E}, whose proof takes over more than 50 pages in the Appendix, as it is a case-by-case analysis of a series of algebras associated to each generalized Dynkin diagram in the classification of matrices $\mathfrak{q}$ from \cite{H-classif}.

Although being tedious and of considerable length (relative to the main body of the paper), we decided to include the full proof here, as there is no general argument underlying it, and each example is analyzed with a subtle combination of different ad-hoc techniques, which are developed in the present article.

\subsection{Organization} In Section \ref{sec:prelim} we review some general Hopf-related concepts and the generic setup for the strategy in \cite{AAGMV}. In \S\ref{sec:cocycles} we present a series of lemmas which set the ground to prove our key Theorem \ref{thm:E}, from which Theorem \ref{thm:cocycle-defs} follows. As stated, the proof of Theorem \ref{thm:E} is completed in the Appendix. In \S\ref{sec:non-connected} we show that the non-connected diagrams can be easily understood from the connected case. Finally, in Section \ref{sec:isos} we study the isomorphism classes of the algebras $\mathfrak{u}(\boldsymbol\lambda)$ from the classification.

\section*{Acknowledgments} We thank Nicol\'as Andruskiewitsch for his constant support and council.
We also thank Cristian Vay for pointing to us a mistake in a previous version of this article.

\section{Preliminaries}\label{sec:prelim}
We work over an algebraically closed field of characteristic zero $\Bbbk$. If $A$ is a $\Bbbk$-algebra and $S\subseteq A$ is a set, then we denote by $\langle S\rangle$ the ideal generated by $S$ in $A$. Idem for a group $G$ and a subset $S\subseteq G$, mutatis mutandis. We denote by $\Alg(A,\Bbbk)$ the set of algebra maps $A\to\Bbbk$.

If $\Gamma$ is an abelian group, then we denote by $\widehat{\Gamma}$ the group of characters of $\Gamma$.  If $n\in \mathbb{N}$, then we set $\I_n\coloneqq\{1,\dots,n\}$, $\I_n^{\circ}\coloneqq\{0,1,\dots,n\}$.

If $H$ is a Hopf algebra, then we denote by $(H_n)_{n\geq 0}$ the coradical filtration of $H$ and set $\gr H=\oplus_{n\geq 0}H_n/H_{n-1}$ the associated graded coalgebra; here $H_{-1}\coloneqq0$.  We denote by $\mathcal{P}(H)$ the subspace of primitive elements in $H$.
If $H'$ is another Hopf algebra, then $\Isom(H,H')$ denotes the set of Hopf algebra isomorphisms $H\to H'$.
We write ${}_{H}\mathcal{M}$ for the category of $H$-modules; an $H$-module algebra is thus an algebra in ${}_{H}\mathcal{M}$.
In turn, $\mathfrak{YD-}$ stands for the category of Yetter-Drinfeld modules over $H$. Given $V\in\mathfrak{YD-}$, $g\in G(H)$ and $\chi\in\Alg(H,\Bbbk)$, then we set
\begin{align}\label{eqn:components}
V_g&\coloneqq\{x\in V| \delta(x)= g\otimes x\}, & V^\chi&\coloneqq\{x\in V| h\cdot x=\chi(h)x,\, h\in H\}.
\end{align}

 If $V\in\mathfrak{YD-}$, then we denote by $\mathfrak{B}(V)$ the {\it Nichols algebra} of $V$.
A {\it lifting} of $V\in\mathfrak{YD-}$ is a Hopf algebra $L$ such that $\gr L\simeq \mathfrak{B}(V)\#H$. A {\it lifting map} is a Hopf algebra projection $\phi:T(V)\#H\twoheadrightarrow L$ such that
\begin{align}\label{eqn:lifting-map}
\phi_{|H}&=\id_H & \text{and} && \phi_{|V\#H}&\colon V\#H\stackrel{\simeq}{\longrightarrow} L_1 &\text{as  Hopf bimodules over }H.
\end{align}

Let $H,H'$ be two Hopf algebras. Then $H'$ {\it is a cocycle deformation of} $H$ if there a is a Hopf cocycle $\sigma:H\to H\otimes \Bbbk$ in such a way that $H'\simeq H_{\sigma}$, the Hopf algebra obtained by deforming the multiplication of $H$. We recall that this is equivalent to the existence of a $(H,H')$-bicleft object $A$. This is the approach we follow in the series. See \cite{S} for details.

\subsection{Diagonal braidings}\label{sec:diagonal}
Let $(V,c)$ be a braided vector space. We set $\theta=\dim V$, $\I=\I_\theta$. Assume that $V$ is of diagonal type, with braiding matrix $\mathfrak{q}=(q_{ij})_{i,j\in\I}$; i.e.~there is a basis $\{x_1,\dots,x_\theta\}$ such that $c=c^\mathfrak{q}$ satisfies:
\[
c^\mathfrak{q}(x_i\otimes x_j)=q_{ij}x_j\otimes x_i, \quad i,j\in\I.
\]
Let ${\mathcal D}$ be the generalized Dynkin diagram associated to $V$ as in \cite{H-classif} and let ${\mathcal D}=\bigsqcup_{i=1}^m {\mathcal D}_i$ be the decomposition of ${\mathcal D}$ into $m$ connected components, $m\in\mathbb{N}$. Given two vertices $i,j\in\I$ of ${\mathcal D}$, then we write $i\sim j$ if they belong to the same connected component and $i\not\sim j$ otherwise. If $V=\bigoplus_{i=1}^m V_i$ is the corresponding decomposition of $V$ into sub-braided vector spaces $V_i$, associated to the connected diagram ${\mathcal D}_i$, $i\in\I_m$, then we refer to each $V_i$, $i\in \I_m$, as a {\it (connected) component} of $V$. We shall denote by
\begin{equation}\label{eqn:G(0)}
{\mathcal G}(0)\coloneqq\{x_ix_j-q_{ij}x_jx_i| i<j \text{ and }i\not\sim j\}
\end{equation}
the set of $q$-commutators of vertices in different components.

We denote by ${\mathcal G}$ a set of generators of the ideal defining the Nichols algebra $\mathfrak{B}_\mathfrak{q}$.
If every connected component of $V$ belongs to the list of \cite{H-classif}, then we take ${\mathcal G}$ as computed in \cite{A-nichols} for each component, union ${\mathcal G}(0)$.

Let $H$ be a Hopf algebra. A principal YD-realization of $V$ over $H$ is a family $(\chi_i,g_i)_{i\in\I}\in \Alg(H,\Bbbk)\times G(H)$ such that $\chi_i(g_j)=q_{ji}, i,j\in \I$, and $\chi_i(h)g_i=\chi_i(h\_{2})h\_{1}g_i{\mathcal S}(h\_{3})$ for all $h\in H$, $i\in\I$. In particular, it follows that $\Gamma=\langle g_1,\dots,g_\theta\rangle<Z(G(H))$. These data realize $(V,c^\mathfrak{q})$ as an object in $\mathfrak{YD-}$ in such a way that $c^\mathfrak{q}$ coincides with the categorical braiding of $\mathfrak{YD-}$ and $x_i\in V_{g_i}^{\chi_i}=V_{g_i}\cap V^{\chi_i}$, $i\in\I$,  cf.~\eqref{eqn:components}.

\subsection{The strategy}\label{sec:strategy}

We recall the setting of a generic step in the algorithm behind our strategy to compute the liftings as cocycle deformations. We refer the reader to \cite{AAGMV,AAG} for details. We fix
\begin{itemize} [leftmargin=*]\renewcommand{\labelitemi}{$\circ$}
\item A cosemisimple Hopf algebra $H$.
\item  $V\in\mathfrak{YD-}$ such that the ideal ${\mathcal J}(V)$ defining the Nichols algebra $\mathfrak{B}(V)$ is generated by a finite set ${\mathcal G}$.
\end{itemize}

Let ${\mathcal G}={\mathcal G}_0\sqcup\dots \sqcup{\mathcal G}_\ell$ be an {\it adapted stratification} of ${\mathcal G}$ cf.~\cite[\S 5.1]{AAGMV}; in particular ${\mathcal G}_k$ is a basis of a Yetter-Drinfeld submodule $\Mc_{k}\subset \mathcal{P}(\mathfrak{B}_k)$, for each\footnote{${\mathcal G}_\ell$ can be chosen so that it is not properly contained in  $\mathcal{P}(\mathfrak{B}_\ell)$, see loc.cit.} $k$.
We set $\mathfrak{B}_0=T(V)$ and denote by $\mathfrak{B}_k=\mathfrak{B}_{k-1}/\langle {\mathcal G}_{k-1}\rangle$, $k\in \I_{\ell+1}$ the corresponding pre-Nichols algebras; also $\mathcal{H}_k=\mathfrak{B}_k\# H$. In loc.~cit.~we introduced an algorithm to produce at each step $k=0,\dots,\ell$ a collection $\Cleft'(\mathcal{H}_k)$ of cleft objects $\mathcal{A}_k$ in such a way that the Schauenburg left Hopf algebra $\mathcal{L}_k$ associated to each pair $(\mathcal{H}_k,\mathcal{A}_k)$, see \cite{S}, satisfies $\gr \mathcal{L}_k\simeq \mathcal{H}_k$; moreover this algebra is a cocycle deformation of $\mathcal{H}_k$ by construction.

In this article we show that a technical condition we introduced in \cite{AAG} for the diagonal case is always satisfied, which shows that the family of Hopf algebras resulting from the $\ell+1$th step is a complete family of liftings of $V$.

The algorithm starts with $\mathfrak{B}_0=T(V)$ and $\Cleft'(\mathcal{H}_0)$ is the singleton $\{\mathcal{A}_0=T(V)\#H\}$, with coaction $\rho_0=\Delta_{\mathcal{T}}$; thus $\mathcal{L}_0=T(V)\#H$. At a given step $k$ we compute, for each $\mathcal{A}_{k-1}\in \Cleft'(\mathcal{H}_{k-1})$ a collection of quotients $\mathcal{A}_{k-1}\twoheadrightarrow \mathcal{A}_k$.
This setup is  depicted in the following {\it snapshot}
from \cite[p. 696]{AAGMV}:
\begin{equation}\label{eqn:snapshot}
\xymatrix{
	\mathcal{T}\ar@{->}[rr]^{\gamma_0=\id}\ar@{->>}[dd]^{\pi_{k-1}}\ar@/^-1.5pc/@{->>}[ddd]_{\pi_{k}}
	^{\,\equiv} &&
	\mathcal{T}\ar@{->>}[dd]_{\tau_{k-1}}\ar@/^1.5pc/@{->>}[ddd]^{\tau_{k}}_{\equiv\,}\ar@{~>}[rr]
	&&L(\mathcal{T},\mathcal{T})\simeq \mathcal{T}\ar@{->>}[dd]\ar@/^3.5pc/@{->>}[ddd]_{\equiv\quad}\\
	&&&&&\\
	\mathcal{H}_{k-1}\ar@{->}[rr]^{\gamma_{k-1}}\ar@{->>}[d]^{\pi'_{k}} &&
	\mathcal{A}_{k-1}\ar@{->>}[d]_{\tau'_{k}}\ar@{~>}[rr]&&L(\mathcal{A}_{k-1},\mathcal{H}_{k-1})\ar@{->>}[d]\\
	\mathcal{H}_{k}\ar@{->}[rr]^{\gamma_{k}} && \mathcal{A}_{k}\ar@{~>}[rr]&&L(\mathcal{A}_{k},\mathcal{H}_{k})
}
\end{equation}
The maps $\gamma_{k-1}$, $\gamma_k$ denote the corresponding sections, and the vertical arrows represent
the (Hopf) algebra projections. The coaction $\rho_k\colon\mathcal{A}_k\to \mathcal{A}_k\otimes \mathcal{H}_k$ is given by
$\rho_k=(\tau_k'\otimes\pi_k')\rho_{k-1}=(\tau_k \otimes\pi_k)\Delta_{\mathcal{T}}$ and the section satisfies $\gamma_k(xh)=\gamma_k(x)h$, $x\in\mathfrak{B}_k$, $h\in H$.

In the final step $k=\ell+1$, $\mathcal{H}_{\ell+1}=\mathfrak{B}(V)\#H$ and $\mathcal{L}=L(\mathcal{A},\mathfrak{B}(V)\#H)$ is a cocycle deformation of $\mathfrak{B}(V)\#H$ satisfying $\gr \mathcal{L}\simeq \mathfrak{B}(V)\# H$, that is a lifting of $V$ for every $\mathcal{A}\in\Cleft'(\mathfrak{B}(V)\#H)$.

\subsection{On the computation of $\mathcal{A}_{k}$}
By \cite[Proposition 5.8]{AAGMV}, each algebra $\mathcal{A}_{k-1}$ can be decomposed as $\mathcal{A}_{k-1}=\mathcal{E}_{k-1}\# H$, where $\mathcal{E}_0=T(V)$ and each $\mathcal{E}_{k-1}$ is an $H$-module algebra. Hence the problem of computing all quotients $\mathcal{A}_{k-1}\twoheadrightarrow\mathcal{A}_k$ translates into that of finding all suitable quotients $\mathcal{E}_{k-1}\twoheadrightarrow \mathcal{E}_k$.

For each $\boldsymbol\lambda=(\lambda_r)_{r\in{\mathcal G}}\in\Bbbk^{{\mathcal G}}$, $\mathcal{E}_k(\boldsymbol\lambda)$ is defined, as a $\Bbbk$-linear space, via
\begin{equation}\label{eqn:E(lambda)}
\mathcal{E}_k=\mathcal{E}_{k}(\boldsymbol\lambda)=\mathcal{E}_{k-1}/\langle \gamma_{k-1}(r)-\lambda_r : r\in {\mathcal G}_{k-1}\rangle,
\end{equation}
If $\mathcal{E}_{k}(\boldsymbol\lambda)\neq 0$, then it is a $\Bbbk$-algebra. If, additionally, $\mathcal{E}_{k}(\boldsymbol\lambda)$ is an $H$-module
algebra, then $\mathcal{A}_k(\boldsymbol\lambda)\coloneqq\mathcal{E}_{k}(\boldsymbol\lambda)\#H\in\Cleft(\mathcal{H}_k)$, by \cite[Proposition 5.8]{AAGMV}.

The next lemma indicates when $\mathcal{E}_k(\boldsymbol\lambda)$ is an $H$-module algebra. In \S \ref{sec:cocycles} we study when $\mathcal{E}_{k}(\boldsymbol\lambda)\neq 0$ for the case of diagonal braidings.

We introduce some terminology first. For each $r\in{\mathcal G}_{k}$ and each $h\in H$, we write the $H$-action as $h\cdot r=\sum_{s\in{\mathcal G}_{k}}\mu_{rs}(h)s$, some $(\mu_{rs}(h))_{s\in{\mathcal G}_{k}}\in\Bbbk$.

\begin{lem}\label{lem:ideal}
Assume that $\mathcal{E}_k(\boldsymbol\lambda)$ is an $H$-module algebra and that the section $\gamma_{k}\colon \mathcal{H}_{k}\to \mathcal{A}_{k}(\boldsymbol\lambda)$ is $H$-linear.
If $\mathcal{E}_{k+1}(\boldsymbol\lambda)\neq 0$, then it is an $H$-module algebra if and only if
\begin{align}\label{eqn:condition-ideal}
\epsilon(h)\lambda_r=\sum\limits_{s\in{\mathcal G}_{k-1}}\mu_{rs}(h)\lambda_s, \qquad \forall\,r\in{\mathcal G}_{k},h\in H.
\end{align}
\end{lem}
\begin{proof} If \eqref{eqn:condition-ideal} holds, then the ideal
$
\langle \gamma_{k}(r)-\lambda_r : r\in {\mathcal G}_{k}\rangle\subset \mathcal{E}_{k}
$
is an $H$-submodule of $\mathcal{E}_{k}$ and thus $\mathcal{E}_{k+1}(\boldsymbol\lambda)$ inherits an $H$-module structure. Conversely,  set $\kappa(h,r)=\sum\limits_{s\in{\mathcal G}_{k-1}}\mu_{rs}(h)\lambda_s-\epsilon(h)\lambda_r\in\Bbbk$, for $h\in H$, $r\in {\mathcal G}_k$. Then
\[
h\cdot (\gamma_k(r)-\lambda_r)=\sum_{s\in{\mathcal G}_{k-1}}\mu_{rs}(h)\left(\gamma(s)-\lambda_s\right) + \kappa(h,r).
\]
Since $\mathcal{E}_{k+1}(\boldsymbol\lambda)\neq 0$, $\kappa(h,r)=0$ for all $h,r$ and  \eqref{eqn:condition-ideal} holds.
\end{proof}

\begin{rem}
If $k=0$, then $\gamma_k=\id$ thus it is $H$-linear. On the other hand, if $H$ is semisimple, then $\gamma_k$ is $H$-linear for every $k$, \cite[Proposition 5.8(c)]{AAGMV}. This is also the case when the braiding is diagonal, see \S \ref{sec:cocycles}.
\end{rem}

\section{All liftings are cocycle deformations}\label{sec:cocycles}

In this section we further assume that
\begin{itemize} [leftmargin=*]\renewcommand{\labelitemi}{$\circ$}
\item
$V\in\mathfrak{YD-}$ comes from a principal realization $(g_i,\chi_i)_{i\in\I_{\theta}}$ of a braided vector space of diagonal type $(V,c)$ with matrix ${\mathfrak q}$.
\end{itemize}  We show that every lifting of $V$ is a cocycle deformation of $\mathfrak{B}_{\mathfrak{q}}\# H$, by showing that the algebras $\mathcal{E}_k(\boldsymbol\lambda)\in{}_{H}\mathcal{M}$ in \eqref{eqn:E(lambda)} are nonzero.

We consider ${\mathcal G}$ as in \S \ref{sec:diagonal}. We set $\Gamma=\langle g_1,\dots,g_\theta\rangle < G(H)$, so each $r\in{\mathcal G}$ belongs to a unique component $r\in T(V)_{g_r}^{\chi_r}$, for some  $g_r\in\Gamma$ and $\chi_r\in\Alg(H,\Bbbk)$, cf.~\eqref{eqn:gr}. We fix an stratification ${\mathcal G}_0\sqcup\dots \sqcup{\mathcal G}_\ell$ in such a way that ${\mathcal G}_0={\mathcal G}(0)$ as in \eqref{eqn:G(0)} and such that $\Bbbk\langle{\mathcal G}_\ell\rangle$ is the normal braided Hopf subalgebra of $\mathfrak{B}_\ell$ generated by the powers of the Cartan root vectors, cf.~\cite[Theorem 31]{A-distinguished}. When $\mathcal{E}_k(\boldsymbol\lambda)\neq 0$ for every $k$, we shall denote
\begin{align}\label{eqn:Es}
\widetilde{\mE}_{\bq}(\boldsymbol\lambda)&\coloneqq \mathcal{E}_{\ell}(\boldsymbol\lambda),& \mathcal{E}_{\mathfrak{q}}(\boldsymbol\lambda)&\coloneqq \mathcal{E}_{\ell+1}(\boldsymbol\lambda).
\end{align}

\smallbreak

In this context, Lemma \ref{lem:ideal} is equivalent to the following. Recall the notation from \eqref{eqn:snapshot}.
\begin{lem}\label{lem:decomposition}
Assume that $\mathcal{E}_k(\boldsymbol\lambda)\neq 0$. Then $\gamma:\mathcal{H}_k\to \mathcal{A}_k(\boldsymbol\lambda)$ is $H$-linear and
$\mathcal{E}_k(\boldsymbol\lambda)$ is an $H$-module algebra if and only if
\begin{equation}\label{eqn:condition-lambda-k}
\lambda_r=0 \quad\text{ if }\chi_r\neq \epsilon, \quad \text{for all }r\in{\mathcal G}_{k-1}.
\end{equation}
Let $\boldsymbol\lambda=(\lambda_r)_{r\in{\mathcal G}}\in\Bbbk$ be a family of scalars satisfying \eqref{eqn:condition-lambda-k}. For each $k\geq 0$,
$\mathcal{E}=\mathcal{E}_k(\boldsymbol\lambda)$ decomposes as a sum of $\Gamma$-eigenspaces
$\mathcal{E}=\bigoplus_{\chi\in\widehat{\Gamma}}\mathcal{E}^\chi$.
\end{lem}
\begin{proof}
We proceed by induction on $k$. As $V=\bigoplus_{i\in\I_\theta} V^{\chi_i}$, the $H$-module algebra $\mathcal{E}_0=T(V)$ decomposes as a sum of $\Gamma$-eigenspaces
 $T(V)=\bigoplus_{\chi\in\widehat{\Gamma}}T(V)^\chi$ and $\gamma=\id:\mathcal{H}_0\to\mathcal{A}_0$ is $H$-linear.

Fix $k\geq 0$ for which the statement holds. By definition, the pre-Nichols algebra $\mathfrak{B}_k$ also decomposes as a sum $\mathfrak{B}_k=\bigoplus_{\chi\in\widehat{\Gamma}}\mathfrak{B}_k^\chi$. Let $\tau_k:T(V)\# H\to \mathcal{A}_k$ be the algebra projection; by assumption, this is an $H$-linear map --which restricts to an $H$-linear map $\tau_k:T(V)\to \mathcal{E}_k=\bigoplus_{\chi\in\widehat{\Gamma}}\mathcal{E}^\chi$. Let $x\#t\in\mathcal{H}_k$, $x\in \mathfrak{B}_k^\chi$, some $\chi\in \widehat{\Gamma}$, $t\in H$. As $\gamma(xt)=\gamma(x)t$ cf.~\eqref{eqn:snapshot}, we may assume $t=1$. Now, let $y\in T(V)^\chi$ be such that $\tau_k(y)=\gamma(x)$. Then
\[
h\cdot \gamma(x)=h\cdot \tau_k(y)=\tau_k(h\cdot y)=\chi(h)\tau_k(y)=\chi(h)\gamma(x), \ \forall\,h\in H.
\]
Hence $\gamma:\mathcal{H}_k\to \mathcal{A}_k$ is $H$-linear and the first assertion follows by Lemma \ref{eqn:condition-ideal}, as  \eqref{eqn:condition-ideal} becomes \eqref{eqn:condition-lambda-k} in this setting. This also shows that the ideal defining $\mathcal{E}_{k+1}$, see \eqref{eqn:E(lambda)}, is generated by $\widehat{\Gamma}$-homogeneous elements, and thus the $\widehat{\Gamma}$-graduation descends to $\mathcal{E}_{k+1}$. The lemma follows.
\end{proof}

Next we show that condition \eqref{eqn:condition-lambda-k} is sufficient. Namely, recall the definition of the set $\mathcal{R}\subset\Bbbk^{{\mathcal G}}$ from \eqref{eqn:R}, then we have:

\begin{theorem}\label{thm:E}
$\mathcal{E}_{\mathfrak{q}}(\boldsymbol\lambda)\neq 0$ if and only if $\boldsymbol\lambda\in\mathcal{R}$.
\end{theorem}

To prove this theorem, we need a series of results, see \S\ref{sec:proof} for the proof.

\begin{cor}\label{cor:A}
For every $\boldsymbol\lambda\in\mathcal{R}$, $\mathcal{A}_k(\boldsymbol\lambda)\coloneqq\mathcal{E}_k(\boldsymbol\lambda)\# H$ is a right $\mathfrak{B}_k\#H$-cleft object. In particular,
\[
\mathcal{A}(\boldsymbol\lambda)\coloneqq\mathcal{E}_{\mathfrak{q}}(\boldsymbol\lambda)\# H\in\Cleft(\mathfrak{B}(V)\# H), \quad \forall\,\boldsymbol\lambda\in\mathcal{R}.
\]
\end{cor}
\begin{proof}
Follows by \cite[Proposition 5.8]{AAGMV}.
\end{proof}

The following result will be useful.

\begin{lem}\label{lem:nonzero-k<l}
Let $k\leq \ell$, $\boldsymbol\lambda\in\mathcal{R}$. Assume that the following conditions hold:
\begin{align*}
\mathcal{E}_{k-1}/\langle \gamma_{k-1}(r) : r\in {\mathcal G}_{k-1}\rangle&\neq 0, & \mathfrak{B}_{k-1}/\langle r-\lambda_r : r\in {\mathcal G}_{k-1}\rangle&\neq 0.
\end{align*}
Then  $\mathcal{E}_k(\boldsymbol\lambda)\neq 0$.
\end{lem}
\begin{proof}
We consider the algebra map $\varrho:\mathcal{E}_{k-1}\to \mathcal{E}_{k-1}\otimes \mathfrak{B}_{k-1}$ given by the braided coaction as in \cite[Lemma 4.1]{AG}. As ${\mathcal G}_{k-1}\subset \mathcal{P}(\mathfrak{B}_{k-1})$,
\[\varrho(\gamma_{k-1}(r))=\gamma_{k-1}(r)\otimes 1+1\otimes r, \quad r\in{\mathcal G}_{k-1}.\]
Now $\varrho$ induces an algebra map
\[
\varrho':\mathcal{E}_{k-1}\to \mathcal{E}_{k-1}/\langle \gamma_{k-1}(r) : r\in {\mathcal G}_{k-1}\rangle \otimes \mathfrak{B}_{k-1}/\langle r-\lambda_r : r\in {\mathcal G}_{k-1}\rangle
\]
such that $\varrho'(\gamma_{k-1}(r)-\lambda_r)=0$, $r\in{\mathcal G}_{k-1}$. Hence $\mathcal{E}_k(\boldsymbol\lambda)\neq 0$.
\end{proof}

\begin{lem}\label{lem:nonzero-k=l+1}
If $\widetilde{\mE}_{\bq}(\boldsymbol\lambda)\coloneqq \mathcal{E}_\ell(\boldsymbol\lambda)\neq 0$, then $\mathcal{E}_{\ell+1}(\boldsymbol\lambda)\neq 0$.
\end{lem}
\begin{proof}
We recall that the normal Hopf subalgebra $\Bbbk\langle{\mathcal G}_{\ell}\rangle\subset \mathfrak{B}_\ell$ is a $q$-poly\-nomial algebra \cite[Proposition 21]{A-distinguished}. Hence, by \cite[Theorem 3.1]{AAGMV}, also \cite[Theorem 4]{Gu}, $\mathcal{E}_{\ell+1}(\boldsymbol\lambda)\neq 0$.
\end{proof}

We fix ${\mathcal D}=\bigsqcup_{i=1}^m {\mathcal D}_i$ and respectively $V=\bigoplus_{i=1}^m V_i$ the decompositions of ${\mathcal D}$  and $V$ into connected components.
We denote by ${\mathcal G}(i)$ the set of generators of the ideal defining the Nichols algebra $\mathfrak{B}(V_i)$, $i\in \I_m$. As a result, the set ${\mathcal G}$ of generators of ${\mathcal J}(V)$, see \S \ref{sec:diagonal}, decomposes as a disjoint union, cf.~\eqref{eqn:G(0)}:
\begin{equation}\label{eqn:Gc=union}
{\mathcal G}={\mathcal G}(0)\sqcup{\mathcal G}(1)\sqcup \dots \sqcup {\mathcal G}(m).
\end{equation}

Let $\mathcal{R}\subseteq\Bbbk^{{\mathcal G}}$, resp.~$\mathcal{R}(i)\subseteq \Bbbk^{{\mathcal G}(i)}$, $i\in \I_m$,  be the set of deformation parameters for $V$, resp.~$V_i$, cf. \eqref{eqn:R}. Then we have a restriction map:
\begin{equation}\label{eqn:restriction}
\mathcal{R}\to \mathcal{R}(i), \qquad \boldsymbol\lambda\mapsto \boldsymbol\lambda_{|i}\coloneqq (\lambda_r)_{r\in{\mathcal G}(i)}\in\mathcal{R}(i).
\end{equation}
In particular, $\boldsymbol\lambda$ splits into a product of subfamilies
\begin{align}\label{eqn:lambda-decomp}
\boldsymbol\lambda=\boldsymbol\lambda_{|0}\times \boldsymbol\lambda_{|1}\times\dots\times\boldsymbol\lambda_{|m}
\end{align}
in such a way that
\begin{itemize}
\item[(i)] $\boldsymbol\lambda_{|0}\in\Bbbk^{{\mathcal G}(0)}$ is the subset of deformation parameters linking vertices in different connected components;
\item[(ii)] $\boldsymbol\lambda_{|i}\in\Bbbk^ {{\mathcal G}(i)}$ is the subset of deformation parameters for the $i$th connected component of $V$, $i\in\I_m$.
\end{itemize}

\begin{lem}\label{lem:lambda'}
Let $\boldsymbol\lambda=(\lambda_r)_{r\in{\mathcal G}}\in\Bbbk$ be a family of scalars satisfying \eqref{eqn:condition-lambda-k}.
If $\mathcal{E}(\boldsymbol\lambda_{|i})\neq 0$ for every $i\in\I_m$ and $\boldsymbol\lambda_{|0}={\bf 0}$, then $\widetilde{\mE}_{\bq}(\boldsymbol\lambda)\neq 0$.
\end{lem}
\begin{proof}
Let $\rho:T(V)\to \End \otimes_{t=1}^m \mathcal{E}(\boldsymbol\lambda_{|t})$ be the algebra map defined by
\begin{align*}
\rho(y_k)(\mathbf{y}_1\otimes \dots \otimes \mathbf{y}_m):= g_k\cdot\mathbf{y}_1\otimes \dots \otimes
g_k\cdot\mathbf{y}_{i-1} \otimes y_k\mathbf{y}_i \otimes \mathbf{y}_{i+1}\otimes \dots \otimes \mathbf{y}_m,
\end{align*}
where $\mathbf{y}_t\in \mathcal{E}(\boldsymbol\lambda_{|t})$, $k\in{\mathcal D}_i$, $i\in\I_m$.
We claim that $\rho(y_{k\ell})=0$ for all $k\not\sim \ell$, $k<\ell$. Indeed, if $k\in{\mathcal D}_i$, $\ell\in{\mathcal D}_j$,  $i<j\in\I_m$, then
\begin{align*}
\rho(y_k y_\ell)(\mathbf{y}_1\otimes & \dots \otimes \mathbf{y}_m):= g_k g_\ell \cdot\mathbf{y}_1\otimes \dots \otimes
g_k g_\ell \cdot\mathbf{y}_{i-1} \otimes y_k(g_\ell \cdot \mathbf{y}_i) \\
& \quad \otimes g_\ell \cdot \mathbf{y}_{i+1}\otimes \cdots \otimes
g_\ell \cdot \mathbf{y}_{j-1} \otimes y_\ell \mathbf{y}_j \otimes \mathbf{y}_{j+1}\otimes \dots \otimes \mathbf{y}_m, \\
\rho(y_\ell y_k )(\mathbf{y}_1\otimes & \dots \otimes \mathbf{y}_m):= g_k g_\ell \cdot\mathbf{y}_1\otimes \dots \otimes
g_k g_\ell \cdot\mathbf{y}_{i-1} \otimes g_\ell \cdot( y_k \mathbf{y}_i) \\
& \quad \otimes g_\ell \cdot \mathbf{y}_{i+1}\otimes \cdots \otimes
g_\ell \cdot \mathbf{y}_{j-1} \otimes y_\ell \mathbf{y}_j \otimes \mathbf{y}_{j+1}\otimes \dots \otimes \mathbf{y}_m.
\end{align*}
As $k\not\sim \ell$, $q_{\ell k}q_{k\ell}=1$ and hence
\begin{align*}
g_\ell \cdot( y_k \mathbf{y}_i) & =  ( g_\ell \cdot y_k) (g_\ell \cdot \mathbf{y}_i)
= q_{\ell k} y_k (g_\ell \cdot \mathbf{y}_i) = q_{k\ell}^{-1} y_k (g_\ell \cdot \mathbf{y}_i),
\end{align*}
so $\rho(y_{k\ell})(\mathbf{y}_1\otimes \dots \otimes \mathbf{y}_m)=0$ for all $\mathbf{y}_t\in \mathcal{E}(\boldsymbol\lambda_{|t})$.

Let $r\in{\mathcal G}(p)$. We claim that $\rho(\gamma_{p-1}(r))(\mathbf{y}_1\otimes \dots \otimes \mathbf{y}_m)=0$.
It suffices to consider $\mathbf{y}_t\in \mathcal{E}(\boldsymbol\lambda_{|t})$ such that
$\mathbf{y}_t \in\pi_t(T(V_t)_{{\bf g}_t}^{\boldsymbol\chi_t} )$ for some ${\bf g}_t\in\Gamma$, $\boldsymbol\chi_t\in\widehat{\Gamma}$, where $\pi_t:T(V_t)\to \mathcal{E}(\boldsymbol\lambda_{|t})$ is the canonical projection. For each $k\in{\mathcal D}_i$,
\begin{multline*}
\rho(y_k)(\mathbf{y}_1\otimes \dots \otimes \mathbf{y}_m)= \boldsymbol\chi_1\dots \boldsymbol\chi_{i-1} (g_k)
\mathbf{y}_1\otimes \dots \otimes \mathbf{y}_{i-1} \otimes y_k\mathbf{y}_i \otimes \mathbf{y}_{i+1}\otimes \dots \otimes \mathbf{y}_m \\
= \chi_k^{-1} ({\bf g}_1\dots {\bf g}_{i-1})
\mathbf{y}_1\otimes \dots \otimes \mathbf{y}_{i-1} \otimes y_k\mathbf{y}_i \otimes \mathbf{y}_{i+1}\otimes \dots \otimes \mathbf{y}_m.
\end{multline*}
Since $\gamma_{p-1}(r)\in \mathcal{E}(\boldsymbol\lambda_{|p})^{\chi_r}$, we have
\begin{multline*}
\rho(\gamma_{p-1}(r))(\mathbf{y}_1\otimes \dots \otimes \mathbf{y}_m)= \\
\chi_r^{-1} ({\bf g}_1\dots {\bf g}_{i-1})
\mathbf{y}_1\otimes \dots \otimes \mathbf{y}_{i-1} \otimes \gamma_{p-1}(r) \mathbf{y}_i \otimes \mathbf{y}_{i+1}\otimes \dots \otimes \mathbf{y}_m=0.
\end{multline*}
Hence $\rho$ induces an algebra map $\widetilde{\rho}:\widetilde{\mE}_{\bq}(\boldsymbol\lambda)\to \End \otimes_{t=1}^m \mathcal{E}(\boldsymbol\lambda_{|t})$ and therefore $\widetilde{\mE}_{\bq}(\boldsymbol\lambda)\neq 0$.
\end{proof}

\begin{lem}\label{lem:lambda''}
If $\boldsymbol\lambda_{|i}={\bf 0}$ for all $i>0$, then $\widetilde{\mE}_{\bq}(\boldsymbol\lambda)\neq 0$.
\end{lem}
\begin{proof}
Set $N_j=\ord \chi_j(g_j)$, $y_{jk}\coloneqq y_jy_k-q_{jk}y_ky_j$, $j<k\in\I$. Let $\mathcal{F}(\boldsymbol\lambda_{|0})$ be the quotient of $T(V)$ by the ideal generated by
\begin{align*}
& y_{jk}-\lambda_{jk}, \ j\not\sim k, & & y_{jk}, \ j\sim k, j<k, && y_j^{N_j}, \ j\in \I.
\end{align*}
An easy application of the Diamond Lemma shows that $\mathcal{F}(\boldsymbol\lambda_{|0})\neq 0$.

For each connected component $V_i$, the relations in ${\mathcal G}(i)$ hold in $\widetilde{\mE}_{\bq}$, since $\boldsymbol\lambda_{|i}={\bf 0}$, cf.~\eqref{eqn:E(lambda)}. Recall that the set ${\mathcal G}(i)$ consists of relations $y_j^{N_j}$, for certain $j\in \I_{\theta_i}$, and relations involving at least two letters $y_j, y_k$ contained in the ideal $\langle y_{jk} | j<k\in \I_{\theta_i}\rangle$. Hence $\widetilde{\mE}_{\bq}(\boldsymbol\lambda)$ projects onto $\mathcal{F}(\boldsymbol\lambda_{|0})$.
\end{proof}

\begin{lem}\label{lem:cocycle-defs-nc}
Let $\boldsymbol\lambda=(\lambda_r)_{r\in{\mathcal G}}\in\Bbbk$ be a family of scalars satisfying \eqref{eqn:condition-lambda-k}.
If $\mathcal{E}(\boldsymbol\lambda_{|i})\neq 0$ for every $i\in\I_m$, then $\mathcal{E}(\boldsymbol\lambda)\neq 0$.
\end{lem}
\begin{proof}
The case  $\boldsymbol\lambda_{|0}={\bf 0}$ follows by Lemma \ref{lem:lambda'}, while the case $\boldsymbol\lambda_{|i}={\bf 0}$ for all $i>0$ is Lemma \ref{lem:lambda''}. Hence we  make use of Lemma \ref{lem:nonzero-k<l} to combine both cases and conclude that $\mathcal{E}(\boldsymbol\lambda)\neq 0$ for any $\boldsymbol\lambda$.
\end{proof}

\subsection{Proof of Theorem \ref{thm:E}}\label{sec:proof}

We show that $\mathcal{E}_k(\boldsymbol\lambda)\neq 0$, for every $k\geq 0$.
We may restrict to the connected case by Lemma \ref{lem:cocycle-defs-nc}. Moreover, we just need to check that $\widetilde{\mE}_{\bq}(\boldsymbol\lambda)\neq 0$, by Lemma \ref{lem:nonzero-k=l+1}. This is done in the Appendix, by a case-by-case analysis of all Nichols algebras following the explicit list given in \cite{AA}. We explain how to define $\widetilde{\mE}_{\bq}(\boldsymbol\lambda)$ recursively and show that $\widetilde{\mE}_{\bq}(\boldsymbol\lambda)\neq 0$ for each $\mathfrak{q}$.
\qed

\subsection{The liftings $\mathfrak{u}(\boldsymbol\lambda)$}\label{sec:u(lambda)}
We review \cite[\S 3.1]{AAG} where the liftings are defined.

We fix $\boldsymbol\lambda\in\mathcal{R}$ as in \eqref{eqn:R}: this defines a right $\mathfrak{B}(V)\#H$-cleft object $\mathcal{A}(\boldsymbol\lambda)$, by Corollary \ref{cor:A}. We consider the left Schauenburg Hopf algebra $\mathcal{L}(\boldsymbol\lambda)\coloneqq L(\mathcal{A}(\boldsymbol\lambda),\mathfrak{B}(V)\#H)$. This is a cocycle deformation of $\mathfrak{B}(V)\#H$ for which $\mathcal{A}(\boldsymbol\lambda)$ becomes a $(\mathfrak{B}(V)\#H,\mathcal{L}(\boldsymbol\lambda)$-bicleft object.

We provide a recursive process to define a Hopf algebra $\mathfrak{u}(\boldsymbol\lambda)$ such that $\mathfrak{u}(\boldsymbol\lambda)\simeq \mathcal{L}(\boldsymbol\lambda)$. Our aim is to get a quotient $T(V)\#H\twoheadrightarrow\mathfrak{u}(\boldsymbol\lambda)$, that is, a definition of the liftings by generators and relations.

We will proceed inductively by considering intermediate quotients $\mathfrak{u}_k(\boldsymbol\lambda)\simeq \mathcal{L}_k(\boldsymbol\lambda)\coloneqq L(\mathcal{H}_k,\mathcal{A}_k(\boldsymbol\lambda))$, recall that $\mathcal{H}_k=\mathfrak{B}_k\#H$.

Recall that $\mathcal{E}_0(\boldsymbol\lambda)=T(V)$; hence $\mathcal{A}_0(\boldsymbol\lambda)=T(V)\# H$ and thus we set $\mathfrak{u}_0(\boldsymbol\lambda)=T(V)\#H\simeq \mathcal{L}_0(\boldsymbol\lambda)$.

Fix $k>0$ and assume that $\mathfrak{u}_{k-1}(\boldsymbol\lambda)$ is already defined. In particular, there is a left coaction $\mathcal{A}_{k-1}(\boldsymbol\lambda)\to \mathfrak{u}_{k-1}(\boldsymbol\lambda)\otimes \mathcal{A}_{k-1}(\boldsymbol\lambda)$. Let $\gamma:\mathcal{H}_{k-1}\to\mathcal{A}_{k-1}(\boldsymbol\lambda)$ be the section corresponding to the right coaction $\mathcal{A}_{k-1}(\boldsymbol\lambda)\to \mathcal{A}_{k-1}(\boldsymbol\lambda)\otimes \mathcal{H}_{k-1}$ and set, for each $r \in {\mathcal G}_{k-1}$,
\begin{align*}
\nabla(r) &:=\gamma(r)_{(-1)}\otimes \gamma(r)_{(0)}
 - g_r \otimes \gamma(r).
\end{align*}
\begin{cor}\cite[Corollary 5.12]{AAGMV}
$\nabla(r)\in \mathfrak{u}_{k-1}(\boldsymbol\lambda) \otimes 1\subset \mathfrak{u}_{k-1}(\boldsymbol\lambda) \otimes \mathcal{A}(\boldsymbol\lambda)$.
\end{cor}
Hence this defines an element $ \tilde r\in \mathfrak{u}_{k-1}(\boldsymbol\lambda)$ so that  $\nabla (r) = \tilde r \otimes 1$. We set
\begin{align}\label{eqn:Lk}
\mathfrak{u}_k(\boldsymbol\lambda):= \mathfrak{u}_{k-1}/\langle \tilde
r-\lambda_r(1-g_r): r\in {\mathcal G}_{k-1}\rangle.
\end{align}
By \cite[Proposition 3.3]{AAG} and Corollary \ref{cor:A}, $\mathfrak{u}_k(\boldsymbol\lambda)\simeq L(\mathcal{A}_k(\boldsymbol\lambda),\mathcal{H}_k)$. Thus, $\mathfrak{u}_k(\boldsymbol\lambda)$ is a quotient of $T(V)\#H=\mathcal{L}_0(\boldsymbol\lambda)$, for each $k\geq 0$. We set, cf.~\eqref{eqn:u}:
\[
\mathfrak{u}(\boldsymbol\lambda)\coloneqq \mathfrak{u}_{\ell+1}(\boldsymbol\lambda).
\]
Our Theorem \ref{thm:E} readily implies the main result of the article, which in this context reads as follows.
\begin{theorem}\label{teo:lift}
Let $L$ be a lifting of $V$. Then there is $\boldsymbol\lambda\in\mathcal{R}$ such that $L\simeq \mathfrak{u}(\boldsymbol\lambda)$.
\end{theorem}
\begin{proof}
\cite[Theorem 3.5]{AAG} applies, by Theorem \ref{thm:E}.
\end{proof}

\section{Diagonal case: non-connected diagrams}\label{sec:non-connected}

We fix a pair $(H,V)$ as in \S \ref{sec:diagonal}. We focus on the case in which the generalized Dynkin diagram ${\mathcal D}$  associated to $V$ is not connected. Our main result is Proposition \ref{pro:joint}, which shows that we can restrict to the connected case as the liftings for non-connected diagrams are defined via linking relations \eqref{eqn:linking} for the liftings of the connected components.

We fix $\{x_1,\dots,x_\theta\}$ a basis of $V$ with $x_i\in V_{g_i}^{\chi_i}$, $i\in\I$. In particular, recall that every $r\in{\mathcal G}$ belongs to a component $ T(V)_{g_r}^{\chi_r}$, for some $g_r\in\Gamma$, $\chi_r\in\Alg(H,\Bbbk)$.

We fix ${\mathcal D}=\bigsqcup_{i=1}^m {\mathcal D}_i$ and respectively $V=\bigoplus_{i=1}^m V_i$ the decompositions of ${\mathcal D}$  and $V$ into connected components.
We keep the notation  as in \S \ref{sec:cocycles}. Notice that \eqref{eqn:restriction} associates a lifting $\mathfrak{u}(\boldsymbol\lambda_{|i})$ of $V_i$ to each lifting $\mathfrak{u}(\boldsymbol\lambda)$ of $V$.
For each $i\in \I_m$, we set ${\mathcal J}(\boldsymbol\lambda_{|i})\subset T(V_i)$ as the ideal such that
\[\mathfrak{u}(\boldsymbol\lambda_{|i})=T(V_i)\# H/{\mathcal J}(\boldsymbol\lambda_{|i}).\]
Let ${\mathcal G}(i)={\mathcal G}(i)_0\sqcup \dots \sqcup {\mathcal G}(i)_{\ell(i)}$ be an stratification of the set of defining relations ${\mathcal G}(i)$ of $\mathfrak{B}(V_i)$. Then
\begin{align}\label{eqn:strat-union}
{\mathcal G}={\mathcal G}(1)_0\sqcup \dots \sqcup {\mathcal G}(1)_{\ell(1)}\sqcup\cdots \sqcup{\mathcal G}(m)_0\sqcup \dots \sqcup {\mathcal G}(m)_{\ell(m)}\sqcup{\mathcal G}(0)
\end{align}
is an stratification of ${\mathcal G}$ with $N+1$ steps, $N=r+\ell(1)+\dots+\ell(m)$.

We denote by $\mathfrak{u}_{|i}(\boldsymbol\lambda)$ the subalgebra of $\mathfrak{u}(\boldsymbol\lambda)$ generated by $H\oplus V_i\#H$; this is indeed a Hopf subalgebra.
\begin{lem}\label{lem:subalgebra}
$\mathfrak{u}_{|i}(\boldsymbol\lambda)\simeq \mathfrak{u}(\boldsymbol\lambda_{|i})$ as Hopf algebras.
\end{lem}
\begin{proof}
Let $\phi:T(V)\# H\to \mathfrak{u}(\boldsymbol\lambda)$ be a lifting map, cf.~\S \ref{sec:prelim}. We assume that $i=1$, for short. We consider the map
$\phi_1:T(V_1)\# H\to \mathfrak{u}(\boldsymbol\lambda)$ given by the composition of $\phi$ with the canonical inclusion $T(V_1)\# H\hookrightarrow T(V)\# H$. Using the stratification as in \eqref{eqn:strat-union}, it follows that $\phi_1({\mathcal J}(\boldsymbol\lambda_{|1}))=0$ and thus $\phi_1$ factors through a map $\varphi:\mathfrak{u}(\boldsymbol\lambda_{|1})\to \mathfrak{u}(\boldsymbol\lambda)$.

By construction, the first term $\mathfrak{u}(\boldsymbol\lambda_{|1})_1$ of the coradical filtration of $\mathfrak{u}(\boldsymbol\lambda_{|1})$ is (isomorphic to) $H\oplus V_1\#H$; similarly $\mathfrak{u}(\boldsymbol\lambda)_1\simeq H\oplus V\#H$. Now, the restriction of $\varphi$ to $\mathfrak{u}(\boldsymbol\lambda_{|1})_1$ is the canonical inclusion  $H\oplus V_1\#H\hookrightarrow H\oplus V\#H$ since $\phi$ is a lifting map. Hence $\varphi$ is injective by \cite[Theorem 5.3.1]{Mo}.
\end{proof}

Now we proceed in the opposite direction, and show in Proposition \ref{pro:joint} that we can combine the liftings of each of the $V_i$'s to find (all) liftings of $V$.

We denote by ${\mathcal J}(\boldsymbol\lambda_{>0})$ the ideal of $T(V)\#H $ generated by $\bigcup_{i\in\I_m}{\mathcal J}(\boldsymbol\lambda_{|i})$; here we consider the natural identification $T(V_i)\#H\hookrightarrow T(V)\#H$. We set
\[
\mathfrak{U}(\boldsymbol\lambda)=T(V)\#H/{\mathcal J}(\boldsymbol\lambda_{>0}).
\]

The subfamily $\boldsymbol\lambda_{|0}$ consists of scalars $(\lambda_{i,j})_{i\not\sim j,i<j}$ subject to
\begin{align*}
\lambda_{i,j}=0 \qquad  \text{ if } \qquad \chi_{i}\chi_{j}\neq\epsilon \text{ or } g_ig_j=1,
\end{align*}
cf.~\eqref{eqn:condition-lambda-k}. We consider the Hopf ideal ${\mathcal J}(\boldsymbol\lambda_{|0})\subset \mathfrak{U}(\boldsymbol\lambda)$ generated by
\begin{align}\label{eqn:linking}
 \ad(x_i)(x_j)-\lambda_{i,j}(1-g_ig_j),\qquad i<j,i\not\sim j
\end{align}
and set $\mathfrak{u}'(\boldsymbol\lambda)=\mathfrak{U}(\boldsymbol\lambda)/{\mathcal J}(\boldsymbol\lambda_{|0})$.

\begin{pro}\label{pro:joint}
$\mathfrak{u}'(\boldsymbol\lambda)$ is a lifting of $V$; moreover, $\mathfrak{u}'(\boldsymbol\lambda)\simeq \mathfrak{u}(\boldsymbol\lambda)$.
\end{pro}
\begin{proof}
We fix an stratification as in \eqref{eqn:strat-union}. Then, for each step $k=0,\dots,N$, there is a surjective Hopf algebra map $\mathcal{L}_k\twoheadrightarrow \mathfrak{U}(\boldsymbol\lambda)$, cf.~\eqref{eqn:Lk}; hence there is a Hopf algebra projection $\tau':\mathcal{L}_N\twoheadrightarrow \mathfrak{u}'(\boldsymbol\lambda)$. Now, the final step $\mathcal{L}_{N+1}=\mathfrak{u}(\boldsymbol\lambda)$ is given precisely by the quotient of $\mathcal{L}_N$ by the relations \eqref{eqn:linking}. Therefore, $\tau'$ factors through a map $\tau:\mathfrak{u}(\boldsymbol\lambda)\twoheadrightarrow \mathfrak{u}'(\boldsymbol\lambda)$; we shall denote by
$\tau_1$ the  restriction of $\tau$ to the first term $\mathfrak{u}(\boldsymbol\lambda)_1\simeq H\oplus V\# H$ of the coradical filtration of $\mathfrak{u}(\boldsymbol\lambda)$.

Conversely, we fix a lifting map $\phi:T(V)\# H\twoheadrightarrow \mathfrak{u}(\boldsymbol\lambda)$, cf.~\S \ref{sec:prelim}. Now, by Lemma \ref{lem:subalgebra}, $\phi({\mathcal J}(\boldsymbol\lambda_{|i}))=0$ for each $i\in\I_m$, and thus $\phi$ factors through a Hopf algebra epimorphism $\phi':\mathfrak{U}(\boldsymbol\lambda)\twoheadrightarrow \mathfrak{u}(\boldsymbol\lambda)$. We observe that ${\mathcal G}(0)$ is composed of primitive elements in $T(V)$, hence relations \eqref{eqn:linking} hold in $\mathfrak{u}(\boldsymbol\lambda)$ by definition, see \eqref{eqn:Lk}. Hence $\phi'({\mathcal J}(\boldsymbol\lambda_{|0}))=0$ and this induces a Hopf algebra projection $\varphi:\mathfrak{u}'(\boldsymbol\lambda)\twoheadrightarrow \mathfrak{u}(\boldsymbol\lambda)$ such that $\varphi_1\coloneqq\varphi_{|H\oplus V\#H}$ is injective, by \eqref{eqn:lifting-map}.

It follows that $\tau_1\circ\varphi_1=\varphi_1\circ \tau_1=\id_{H\oplus V\#H}$, hence $\tau:\mathfrak{u}(\boldsymbol\lambda)\to\mathfrak{u}'(\boldsymbol\lambda)$ defines a Hopf algebra isomorphism, with inverse $\varphi$.
\end{proof}

\section{On isomorphism classes}\label{sec:isos}

Let $(V,H)$ be as in \S \ref{sec:diagonal}.  If $\boldsymbol\lambda\in\mathcal{R}$, cf.~\eqref{eqn:R}, then we say that $(H,V,\boldsymbol\lambda)$ is a {\it lifting datum}.
Let $(V',H')$ a new pair as in \S \ref{sec:diagonal}; we denote by $((g'_i,\chi'_i))_{i\in\I_{\theta'}}$, $\theta'=\dim V'$, $\mathcal{R}'$, the corresponding input information.

When $V,V'$ are connected, \cite[Theorem 3.9]{AAG} describes the set of isomorphisms classes $\Isom(\mathfrak{u}(\boldsymbol\lambda),\mathfrak{u}(\boldsymbol\lambda'))$ in terms of certain symmetries of the lifting data.  This is in turn inspired by \cite[Theorem 7.2]{AS-annals}, where this is done for certain families of liftings of Cartan type with minor restrictions on the parameters; see also \cite[Theorem 7.1]{AS} for some families of  type $A$.

The claim of \cite[Theorem 3.9]{AAG} extends verbatim to the non-connected case, see Theorem \ref{thm:isos}; also Remark \ref{rem:hypothesis}.
In addition, we review this case in the spirit of Proposition \ref{pro:joint}, that is we study the isomorphism classes of the algebras $\mathfrak{u}(\boldsymbol\lambda)$ in terms of the components of $V$, see Proposition \ref{pro:iso}.

Consider the  decompositions $V=\bigoplus_{i=1}^m V_i$ and $V'=\bigoplus_{i=1}^{m'} V'_i$ of $V$ and $V'$
as sums of connected braided vector subspaces; set $\theta_i=\dim V_i$, $\theta'_j=\dim V_j'$, $i\in\I_m, j\in \I_{m'}$. We fix $\I=\I_\theta$ and set $\I(i)=\{j\in\I | g_j=g_i\text{ and }\chi_j=\chi_i\}$, for each $i\in \I$ and consider the groups
\begin{align*}
\mathbb{S}_{\mathfrak{q}}&=\{\sigma\in\mathbb{S}_\theta| q_{ij} = q_{\sigma(i) \sigma(j)} \,\forall i, j \in \I \};\\
\mathbf{L}&=\{s\in \GL_\theta(\Bbbk)|s_{ij}=0 \text{ if } j\notin\I(i)\}.
	\end{align*}
These groups act on $\mathcal{R}$ \cite[Lemmas 3.10 \& 3.11]{AAG}; we write each action as
\begin{align}\label{eqn:actions}
\boldsymbol\lambda&\mapsto \boldsymbol\lambda^\sigma, & \boldsymbol\lambda&\mapsto s\cdot \boldsymbol\lambda; & \text{for }\boldsymbol\lambda\in\mathcal{R}, \ \sigma\in \mathbb{S}_{\mathfrak{q}}, \ s\in \mathbf{L}.
\end{align}

\begin{rem}\label{rem:action}
Let $\boldsymbol\lambda\in\mathcal{R}$. Then the actions \eqref{eqn:actions} restrict to each component $\boldsymbol\lambda_{|i}$, $i\geq 0$, as in \eqref{eqn:lambda-decomp}, cf.~\cite[Lemmas 3.10 \& 3.11]{AAG}, also \cite[Example 3.12]{AAG}. In particular,
\begin{align}\label{eqn:actions-0}
(s\cdot \boldsymbol\lambda^\sigma)_{|0}=(s_{ij}^{-1}\lambda_{\sigma^{-1}(i),\sigma^{-1}(j)})_{i,j}, \qquad \sigma\in \mathbb{S}_{\mathfrak{q}}, s\in \mathbf{L}.
\end{align}
Here, if $\boldsymbol\lambda_{|0}=(\lambda_r)_{r\in{\mathcal G}(0)}$, then we set $\lambda_{i,j}\coloneqq\lambda_r$ for $i<j\in\I$ defining the q-commutator $r$ cf.~\eqref{eqn:G(0)}.
\end{rem}

\begin{rem}\label{rem:hypothesis}
When $V$ connected, the sets $\I(i)$ and $\mathbf{L}$ are explicitly described  in \cite[\S 3.2.1]{AAG}. This is indeed the only occurrence of the connectedness hypothesis in loc.cit.
\end{rem}

We recall from \cite[Lemma 3.7]{AAG} that if $\psi\in\Isom(\mathfrak{u}(\boldsymbol\lambda), \mathfrak{u}(\boldsymbol\lambda'))$, then $\varphi\coloneqq\psi_{|H}\in\Isom(H,H')$ and $T\coloneqq \psi_{|V}\colon V\to V'$ is an isomorphism of braided vector spaces; so $\theta=\theta'$, $m=m'$ and there is $s=(s_{ij})\in  \mathbf{L}$ with
\begin{align*}
T(a_i)=\sum\limits_{j\in\I(\sigma(i))} s_{ij}a'_j, \ i\in\I,
\end{align*}
where we denote by $\{a_i\}_{i\in\I}$, resp. $\{a'_i\}_{i\in\I}$, the images of the elements $x_i\in V$, resp. $x_i'\in V'$, $i\in\I$, via the natural projections $T(V)\#H\twoheadrightarrow \mathfrak{u}(\boldsymbol\lambda)$, resp. $T(V')\#H'\twoheadrightarrow \mathfrak{u}(\boldsymbol\lambda')$.
In turn, $\varphi$ defines an element $\sigma\in\mathbb{S}_{\mathfrak{q}}$ by the identities
\begin{align}\label{eqn:sigma}
\varphi(g_{j})&=g'_{\sigma(j)} \quad \text{ and }\quad \chi'_{\sigma(j)}\circ \varphi=\chi_j, \ j\in\I.
\end{align}
These data actually determine the isomorphism $\psi$, see Theorem \ref{thm:isos}. We let $\Isom(\boldsymbol\lambda,\boldsymbol\lambda')$ be the set of all triples $(\varphi,\sigma,s)\in\Isom(H,H')\times \mathbb{S}_{\mathfrak{q}}\times \mathbf{L}$ satisfying \eqref{eqn:sigma} and such that $\boldsymbol\lambda'=s\cdot\boldsymbol\lambda^\sigma$.

\begin{theorem}\cite[Theorem 3.9]{AAG}\label{thm:isos}
Let $(H,V,\boldsymbol\lambda)$ and $(H',V',\boldsymbol\lambda')$ be two lifting data. Then
$\Isom(\mathfrak{u}(\boldsymbol\lambda),\mathfrak{u}(\boldsymbol\lambda'))\simeq \Isom (\boldsymbol\lambda,\boldsymbol\lambda').$\qed
\end{theorem}

\begin{rem}\label{rem:not-connected}
If $s\in\mathbf{L}$ and $i\not\sim j\in\I$, then one of the following holds:
\begin{itemize}
\item[(a)]$s_{ij}=0$;
\item[(b)] $\Bbbk\{x_i\}\simeq\Bbbk\{x_j\}\simeq V_p$ for some $p\in\I_m$ and $q_{ii}=q_{ij}=q_{ji}=q_{jj}=-1$.
\end{itemize}
Indeed, given such $i,j$, if there is $k\neq i,j$ connected with one of them, say $k\sim i$, and $s_{ij}\neq 0$, then $j\in\I(i)$, that is $\chi_j=\chi_i$ and $g_j=g_i$. Now, as $k\not\sim j$:
\[
1=\chi_j(g_k)\chi_k(g_j)=\chi_i(g_k)\chi_k(g_i),
\]
which contradicts the fact that $k\sim i$. Hence $s_{ij}=0$. If, on the contrary, both $x_i$ and $x_j$ determine a one-dimensional connected component and $s_{ij}\neq 0$, then $j\in\I(i)$ implies $q_{ii}=q_{ij}=q_{ji}=q_{jj}=-1$.
\qed
\end{rem}

We recall from p.~\pageref{lem:subalgebra} that $\mathfrak{u}_{|i}(\boldsymbol\lambda)\subseteq \mathfrak{u}(\boldsymbol\lambda)$ stands for the subalgebra generated by $V_i$ and $H$. The following is straightforward.

\begin{lem}\label{lem:iso}
Let $\psi\in\Isom(\mathfrak{u}(\boldsymbol\lambda), \mathfrak{u}(\boldsymbol\lambda'))$. Then  there is $\tau\in\mathbb{S}_m$ such that $T_i\coloneqq T_{|V_i}:V_i\to V'_{\tau(i)}$ is an isomorphism of braided vector spaces for each $i\in \I_m$. Moreover, \[\psi_{|i}\coloneqq \psi_{|\mathfrak{u}_{|i}(\boldsymbol\lambda)}:\mathfrak{u}_{|i}(\boldsymbol\lambda)\to \mathfrak{u}_{|\tau(i)}(\boldsymbol\lambda')\] is a Hopf algebra isomorphism.
\qed
\end{lem}
By Lemma \ref{lem:subalgebra}, $\mathfrak{u}_{|i}(\boldsymbol\lambda)\simeq \mathfrak{u}(\boldsymbol\lambda_{|i})$ and hence
each $\psi\in\Isom(\mathfrak{u}(\boldsymbol\lambda), \mathfrak{u}(\boldsymbol\lambda'))$ induces isomorphisms $\mathfrak{u}(\boldsymbol\lambda_{|i})\to \mathfrak{u}(\boldsymbol\lambda_{|\tau(i)}')$ for all $i\in \I_m$. We denote by
\[\Isom(\boldsymbol\lambda_{|1},\boldsymbol\lambda_{|\tau(1)}')\times_H \dots\times_H \Isom(\boldsymbol\lambda_{|m},\boldsymbol\lambda_{|\tau(m)}')\subset \bigtimes_{i=1}^m\Isom(\boldsymbol\lambda_{|i},\boldsymbol\lambda_{|\tau(i)}')
\]
the subset of all $m$-tuples $\left((\varphi_i,\sigma_i,s_i)\right)_{i\in\I_m}$ with $\varphi_i=\varphi_j$, $i,j\in\I_m$. Therefore, Lemma \ref{lem:iso} defines a map
\begin{align}\label{eqn:times-H}
\Res\colon\Isom(\boldsymbol\lambda,\boldsymbol\lambda')&\longrightarrow \Isom(\boldsymbol\lambda_{|1},\boldsymbol\lambda_{|\tau(1)}')\times_H \dots\times_H \Isom(\boldsymbol\lambda_{|m},\boldsymbol\lambda_{|\tau(m)}')
\end{align}

\begin{lem}\label{lem:iso-inj}
The map $\Res$ is injective.
\end{lem}
\begin{proof}
Let $(\varphi,\sigma,s)\in \Isom(\boldsymbol\lambda,\boldsymbol\lambda')$ and denote by $\psi:\mathfrak{u}(\boldsymbol\lambda)\to \mathfrak{u}(\boldsymbol\lambda')$ the corresponding Hopf algebra isomorphism from Theorem \ref{thm:isos}.
The collection of maps $(\psi_{|i})_{i\in\I_m}$ controls the behavior of $\psi$ when restricted to the first term of the coradical filtration of $\mathfrak{u}(\boldsymbol\lambda)$ and hence determines the morphism $\psi$.
\end{proof}
To describe the image of the map $\Res$, we set, for each $i\in\I_m$,
\[
\Isom_0(\boldsymbol\lambda_{|i},\boldsymbol\lambda_{|\tau(i)}')=\{(\varphi,\sigma,s)\in \Isom(\boldsymbol\lambda_{|i},\boldsymbol\lambda_{|\tau(i)}')|\lambda_0'=s\cdot \lambda_0^{\sigma_i}\}
\]
and define $\Isom_0(\boldsymbol\lambda_{|1},\boldsymbol\lambda_{|\tau(1)}')\times_H \dots\times_H \Isom_0(\boldsymbol\lambda_{|m},\boldsymbol\lambda_{|\tau(m)}')$ accordingly.
By definition, the image of $\Res$ lies inside this subset.
\begin{pro}\label{pro:iso}
The map $\Res$ defines a bijective correspondence
\begin{align}\label{eqn:times-H-0}
\Isom(\boldsymbol\lambda,\boldsymbol\lambda')\overset{\sim}{\longrightarrow}\Isom_0(\boldsymbol\lambda_{|1},\boldsymbol\lambda_{|\tau(1)}')\times_H \dots\times_H \Isom_0(\boldsymbol\lambda_{|m},\boldsymbol\lambda_{|\tau(m)}').
\end{align}
\end{pro}
\begin{proof}
First, $\Res$ is injective by Lemma \ref{lem:iso-inj}. Now, let $\left((\varphi,\sigma_i,s_i)\right)_{i\in\I_m}$ belong to the right hand side of \eqref{eqn:times-H-0}, for some  $\varphi\in\Isom(H,H')$. Then, via the natural identification $\I_{\theta_1}\times \dots \times \I_{\theta_m}=\I_\theta$:
\begin{itemize}
\item[(a)] $\sigma=\sigma_1\times\dots\times\sigma_m\in\mathbb{S}_{\mathfrak{q}}$.
\item[(b)] The block matrix
$
s=\left[\begin{smallmatrix}
s_1 & 0 & \cdots &0 \\
0& s_2 & & 0  \\
0 & 0 & \ddots & \vdots \\
0 & 0 & \cdots  & s_m
\end{smallmatrix}\right]$ defines an element in $\mathbf{L}$.

\end{itemize}
Indeed, (a) is clear from the definition and (b) follows by Remark \ref{rem:not-connected}. Hence, $(\varphi,\sigma,s)\in\Isom (\boldsymbol\lambda,\boldsymbol\lambda')$ and it is mapped onto $\left((\varphi,\sigma_i,s_i)\right)_{i\in\I_m}$ by $\Res$.
\end{proof}

\section*{Appendix: nonzero cleft objects}

In this section we show that each algebra $\widetilde{\mathcal E}_{\bq}=\widetilde{\mathcal E}_{\bq}(\boldsymbol\lambda)$ is nonzero for each connected matrix ${\mathfrak q}$ and each $\boldsymbol\lambda\in\mathcal{R}$, see  \eqref{eqn:R}. This completes the proof of Theorem \ref{thm:E}.
We refer to \cite[\S 2.2]{AA} for the notation $x_{ij}$, etc. We use the corresponding notation $y_{ij}$, etc., for the analogous elements in terms of the generators $y_i$, $i\in\I$.

\input{appendix-tools}

\input{appendix}

redpajama_set_name: RedPajamaArXiv