text:
\section{Introduction}\label{introduction}

The dihedral Lie algebra  $D(\Gamma)_{\bullet\bullet}$ is the bigraded Lie algebra introduced in \cite{G}
which is associated with a finite abelian group $\Gamma$.
It reflects the double shuffle and distribution relations among multiple polylogarithms evaluated at roots of unity.
Its relation with a certain bigraded variant of  motivic Lie algebra is discussed in ~loc.~cit.

Kashiwara-Vergne Lie algebra $\mathfrak{krv}_\bullet$ is the filtered graded Lie algebra
introduced in  \cite{AT} and \cite{AET}. It acts on the set of solutions of
\lq a formal version' of Kashiwara-Vergne conjecture.
Related to conjectures on mixed Tate motives,  it is
expected to be isomorphic to the motivic Lie algebra (cf. \cite{F}).

A bigraded variant $\mathfrak{lkrv}_{\bullet\bullet}$ of $\mathfrak{krv}_\bullet$
is introduced
and its mould theoretical interpretation \cite{E81} is investigated
in \cite{RS}.
The results in \cite{M, RS} give  an inclusion of bigraded Lie algebras
\begin{equation}\label{D to lkrv}
D(\{e\})_{\bullet\bullet}\hookrightarrow\mathfrak{lkrv}_{\bullet\bullet}.
\end{equation}
Our objective of  this paper is to
extend it
to any $\Gamma$
by exploiting Ecalle's mould theory (\cite{E-ARIGARI, E-flex})
with self-contained proofs.
Our results are exhibited as follows:

\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item
In Definition \ref{defn: KV condition},
we introduce the filtered graded $\mathbb Q$-linear space $\mathfrak{krv}(\Gamma)_\bullet$
which generalizes $\mathfrak{krv}_{\bullet}$.
In Theorem \ref{thm:reform:krv}, we show that
$\mathfrak{krv}(\Gamma)_\bullet$ is identified with the $\mathbb Q$-linear space of finite polynomial-valued alternal moulds
satisfying Ecalle's senary relation \eqref{senary relation}
and whose $\mathrm{swap}$'s \eqref{eq:swap} are $\mathrm{pus}$-neutral \eqref{pus-neutral},
that is, there is an isomorphism of $\mathbb Q$-linear spaces
$
\mathfrak{krv}(\Gamma)_\bullet\sime
\mathrm{ARI}(\Gamma)_{\mathrm{sena}/\mathrm{pusnu}}\cap\mathrm{ARI}(\Gamma)_\mathrm{al}^{\mathrm{fin},\mathrm{pol}}.$$
\item
In Definition \ref{defn:lrkv},
we introduce a bigraded $\mathbb Q$-linear space $\mathfrak{lkrv}(\Gamma)_{\bullet\bullet}$
which is defined by the \lq leading terms' of the defining equations of
$\mathfrak{krv}(\Gamma)_{\bullet}$.
We also consider its subspace $\mathfrak{lkrvd}(\Gamma)_{\bullet\bullet}$ by imposing the distribution relation in Definition \ref{def:LKRVD}.
Both of them  recover  $\mathfrak{lkrv}_{\bullet\bullet}$ when $\Gamma=\{e\}$.
It is shown  that they form Lie algebras
in  Theorem \ref{thm:lkrv Lie alg}  and Corollary \ref{cor:reform:lkrvd}.
An inclusion of bigraded Lie algebras
\begin{equation*
\mathrm{gr}_{\mathcal{D}}\mathfrak{krv}(\Gamma)_\bullet \hookrightarrow
\mathfrak{lkrv}(\Gamma)_{\bullet\bullet}
\end{equation*}
is presented in \eqref{eq:inclusion grKRV},
where the first term means
the associated bigraded
of the filtered graded linear space $\mathfrak{krv}(\Gamma)_\bullet$.
\item
In Theorem \ref{thm:reform:krv:bigrade}, we show that
$\mathfrak{lkrv}(\Gamma)_{\bullet\bullet}$
is identified with
the Lie algebra (cf. Theorem \ref{thm: ARIpushpusnu Lie}) of finite polynomial-valued alternal moulds
which are $\mathrm{push}$-invariant \eqref{push-invariant}
and whose $\mathrm{swap}$'s are $\mathrm{pus}$-neutral \eqref{pus-neutral},
that is, there is an isomorphism of Lie algebras
$
\mathfrak{lkrv}(\Gamma)_{\bullet\bullet}\simeq\mathrm{ARI}(\Gamma)_{\mathrm{push}/ \mathrm{pusnu}}\cap \mathrm{ARI}(\Gamma)_\mathrm{al}^{\mathrm{fin},\mathrm{pol}}.$$
\item  In \S \ref{sec:dihedral Lie algebra},
we consider Goncharov's dihedral bigraded Lie algebra $D(\Gamma)_{\bullet\bullet}$
and its related Lie algebra ${\mathbb D}(\Gamma)_{\bullet\bullet}$ with the dihedral symmetry  which contains $D(\Gamma)_{\bullet\bullet}$.
It is explained in Theorem \ref{thm:reform:dihedral} that
its depth>1-part of ${\mathbb D}(\Gamma)_{\bullet\bullet}$
coincides with the depth>1-part of the Lie algebra of finite polynomial-valued part of
the set of moulds $\mathrm{ARI}(\Gamma)_{\underline{\mathrm{al}}/\underline{\mathrm{al}}}$ (cf. Definition \ref{def:ARIalal}),
namely
$$
\mathrm{Fil}^2_\mathcal{D} {\mathbb D}(\Gamma)_{\bullet\bullet}\simeq
\mathrm{Fil}^2_\mathcal{D} \mathrm{ARI}(\Gamma)_{\underline{\mathrm{al}}/ \underline{\mathrm{al}}}^{\mathrm{fin},\mathrm{pol}}.
$$
In Theorem \ref{thm:embedding}, we show  that
there is an inclusion of graded Lie algebras
$$
\mathrm{Fil}_{\mathcal{D}}^{2}\mathrm{ARI}(\Gamma)_{\underline{\mathrm{al}}/ \underline{\mathrm{al}}}\hookrightarrow
\mathrm{ARI}(\Gamma)_{\mathrm{push}/ \mathrm{pusnu}}.
$$
In Corollary \ref{cor:embedding}, by taking an intersection with $ \mathrm{ARI}(\Gamma)_\mathrm{al}^{\mathrm{fin},\mathrm{pol}}$
we obtain the inclusion of bigraded Lie algebras from
the depth >1 part $\mathrm{Fil}^2_{D}{\mathbb D}(\Gamma)_{\bullet\bullet}$ of ${\mathbb D}(\Gamma)_{\bullet\bullet}$
to $\mathfrak{lkrv}(\Gamma)_{\bullet\bullet}$
$$
\mathrm{Fil}_{\mathcal{D}}^{2}{\mathbb D}(\Gamma)_{\bullet\bullet}\hookrightarrow\mathfrak{lkrv}(\Gamma)_{\bullet\bullet},
$$
which
extends \eqref{D to lkrv}.
By imposing the distribution relation there
the inclusion
$\mathrm{Fil}_{\mathcal{D}}^{2}{D}(\Gamma)_{\bullet\bullet}\hookrightarrow\mathfrak{lkrvd}(\Gamma)_{\bullet\bullet}$
is similarly obtained in Corollary \ref{cor:embedding2}.
\end{enumerate}

In Appendix \ref{sec:appendix},
we give self-contained proofs of  several
fundamental properties on the $\mathrm{ari}$-bracket of  the Lie algebra $\mathrm{ARI}(\Gamma)$
of moulds associated with $\Gamma$.
In Appendix \ref{sec:MPL at roots},
we discuss moulds arising from the  multiple polylogarithms
evaluated at roots of unity.

\subsubsection*{Acknowledgements}
We thank for L. Schneps who gave comments on the first version of the paper
and informing us \cite{RS}.
H.F. and N.K. have been supported by grants JSPS KAKENHI  JP18H01110
and  JP18J14774 respectively.

\section{Preparation on mould theory}\label{sec:preparation on mould theory}
We prepare several techniques of moulds which will be employed in our later sections.
The notion of moulds, the alternality, flexions and the ari-bracket
associated with a finite group $\Gamma$
are explained
in \S\ref{subsec:moulds} and \S\ref{subsec:ari-bracket}.
In \S\ref{subsec:swap}, we explain that the set
$\mathrm{ARI}(\Gamma)_{\underline\mathrm{al}/\underline\mathrm{al}}$ of
bialternal moulds forms a Lie algebra under the ari-bracket
(whose self-contained proof is given in Appendix \ref{sec:appendix}).
In \S\ref{subsec:push-invariance and pus-neutrality},
we introduce the  set $\mathrm{ARI}_{\mathrm{push}/\mathrm{pusnu}}(\Gamma)$ of push-invariant and pus-neutral
moulds and show that it forms a Lie algebra under the $\mathrm{ari}$-bracket
in Theorem \ref{thm: ARIpushpusnu Lie}.
\subsection{Moulds and alternality}\label{subsec:moulds}
We introduce and discuss moulds associated with a finite abelian group $\Gamma$.

The notion of moulds was invented by Ecalle (cf. \cite[Tome I. pp.12-13]{E81}).
For our convenience we employ the following formulation
influenced by  \cite{S-ARIGARI}
which is different from  the one employed in \cite[D\'{e}finition 1 or D\'{e}finition II.1]{Cre} and \cite[\S 4.1]{Sau}.

Let $\Gamma$ be a finite abelian group.
We set
$\mathcal{F}:=\bigcup_{m\geqslant 1}\mathbb
(x_1,\dots,x_m)$.

\begin{defn
\label{def:mould}
A {\it mould} on $\mathbb{Z}_{\geqslant0}$ with values in $\mathcal{F}$
is  a collection (a sequence)
\begin{equation*}
	M=(M^m(x_1,\dots, x_m))_{m\in\mathbb{Z}_{\geqslant0}}=\bigl(M^0(\emptyset),\ M^1(x_1),\ M^2(x_1, x_2),\ \dots\bigr),
\end{equation*}
with $M^0(\emptyset)\in\mathbb Q$ and $M^m(x_1,\dots,x_m)$
$\in\mathbb Q(x_1,\dots,x_m)^{\oplus\Gamma^{\oplus m}}$ for $m\geqslant 1$,
which is described by a summation
$$
M^m(x_1,\dots, x_m)
\underset{(\sigma_1,\dots,\sigma_m)\in\Gamma^{\oplus m}}{\oplus}
M^m_{\sigma_1, \dots, \sigma_m}(x_1,\ \dots,\ x_m)
$$
where each $M^m_{\sigma_1, \dots, \sigma_m}(x_1,\ \dots,\ x_m)\in\mathbb Q(x_1,\dots,x_m)$.
We denote the set of all moulds with values
in $\mathcal{F}(\Gamma)$ by $\mathcal{M}(\mathcal{F};\Gamma)$.
The set $\mathcal M(\mathcal F;\Gamma)$ forms a $\mathbb Q$-linear space by
\begin{align*}
	A+ B
	&:= (A^m(x_1,\dots, x_m)+ B^m(x_1,\dots, x_m))_{m\in\mathbb{Z}_{\geqslant0}}, \\
	c A
	&:= (c A^m(x_1,\dots, x_m))_{m\in\mathbb{Z}_{\geqslant0}},
\end{align*}
for $A, B\in\mathcal M(\mathcal F;\Gamma)$ and $c\in \mathbb Q$,
namely the addition and the scalar are taken componentwise.
We define a product on $\mathcal M(\mathcal F;\Gamma)$ by
\begin{equation*}
	 (A\times B)^m_{\sigma_1, \dots, \sigma_m}(x_1, \dots, x_m)
	:=\sum_{i=0}^mA^i_{\sigma_1, \dots, \sigma_i}(x_1, \dots, x_{i})
	B^{m-i}_{\sigma_{i+1}, \dots, \sigma_m}(x_{i+1}, \dots, x_m),
\end{equation*}
for $A,B\in\mathcal M(\mathcal F;\Gamma)$ and for $m\geqslant0$ and for $(\sigma_1,\dots,\sigma_m)\in\Gamma^{\oplus m}$.
Then the pair $(\mathcal M(\mathcal F;\Gamma),\times)$ is a non-commutative, associative, unital $\mathbb Q$-algebra. Here, the unit $ I\in\mathcal M(\mathcal F;\Gamma)$ is given by
$ I:=(1,0,0,\dots)$.
\end{defn}

By the regular action of $\Gamma$ on $\mathbb Q[\Gamma]$,
$\mathcal M(\mathcal F;\Gamma)$ admits the action of $\Gamma$.
It is described by
$$(\gamma M)^m_{\sigma_1, \dots, \sigma_m}(x_1,\ \dots,\ x_m)
=M^m_{\gamma^{-1}\sigma_1, \dots, \gamma^{-1}\sigma_m}(x_1,\ \dots,\ x_m)
$$
for $\gamma\in\Gamma$.

The set  $\mathcal M(\mathcal F:\Gamma)$ is encoded with the depth filtration
$\{\mathrm{Fil}^m_{\mathcal{D}}\mathcal M(\mathcal F;\Gamma)\}_{m\geqslant 0}$
where
$\mathrm{Fil}^m_{\mathcal{D}}\mathcal M(\mathcal F;\Gamma)$ is the collection  of moulds
with $ M^r(x_1,\dots,x_r)=0$ for $r<m$.
It is clear that the algebra structure of $\mathcal M(\mathcal F;\Gamma)$ is compatible with the depth filtration.
Put
 $$\mathrm{ARI}(\Gamma):=\{ M\in\mathcal M(\mathcal F;\Gamma)\ |\  M^0(\emptyset)=0\}.$$
It is a filtered (non-unital) subalgebra.

\begin{rem}
Ecalle introduced the notion of bimoulds in \cite[\S 5]{E-ARIGARI} and \cite[\S 1.3]{E-flex}
(cf. \cite[\S 2.1]{S-ARIGARI})
which are the so-called  \lq moulds  with double layered indices'.
We often regard our moulds in Definition \ref{def:mould} as if a bimould and
we sometimes denote $M^m_{\sigma_1, \dots, \sigma_m}(x_1,\ \dots,\ x_m)$ by $M^m
\varia{x_1,\ \dots,\ x_m}{\sigma_1,\ \dots,\ \sigma_m}$ for $M\in\mathrm{ARI}(\Gamma)$
as in ~loc.~cit.
\end{rem}

\begin{defn}
A mould $ M\in \mathcal M(\mathcal F;\Gamma)$ is called {\it finite} when
$ M^m(x_1,\dots, x_m)=0$ except for finitely many $m$.
It is called {\it polynomial-valued}
when $ M^m_{\sigma_1, \dots, \sigma_m}(x_1,\dots, x_m)\in\mathbb Q[x_1,\dots,x_m]$ for all $(\sigma_1, \dots, \sigma_m)\in\Gamma^{\oplus m}$ and $m$.
We denote $\mathcal M(\mathcal F;\Gamma)^{\mathrm{fin},\mathrm{pol}}$
(resp. $\mathrm{ARI}(\Gamma)^{\mathrm{fin},\mathrm{pol}}$)
to be
the subset of all finite polynomial-valued  moulds in $\mathcal M(\mathcal F;\Gamma)$ (resp. $\mathrm{ARI}(\Gamma)$).
\end{defn}

We prepare the following algebraic formulation which is useful to present  the notion of the alternality of mould:
Put $X:=\left\{\binom{x_i}{\sigma}\right\}_{i\in\mathbb{N},\sigma\in\Gamma}$. Let $X_{\mathbb{Z}}$ be the set such that
$$X_{\mathbb{Z}}:=\left\{\varia{u}{\sigma}\ \middle|\ u=a_1x_1+\cdots +a_kx_k,\ k\in\mathbb{N},\ a_j\in\mathbb{Z},\ \sigma\in\Gamma\right\},$$
and let
$X_{\mathbb{Z}}^\bullet$
be the non-commutative free monoid generated by all elements of $X_{\mathbb{Z}}$
with the empty word $\emptyset$ as the unit.
Occasionally we denote each element $\omega=u_1\cdots u_m\in X_{\mathbb{Z}}^\bullet$
with $u_1,\dots,u_m\in X_{\mathbb{Z}}$ by
$\omega=(u_1,\dots,u_m)$
as a sequence.
The {\it length} of $\omega=u_1\cdots u_m$ is defined to be $l(\omega):=m$.

	For our simplicity we  occasionally denote $M\in\mathcal M(\mathcal F;\Gamma)$ by
	\begin{equation*}
		 M=( M^m({\bf x}_m))_{m\in\mathbb{Z}_{\geqslant0}} \quad \text{ or }\quad
		 M=( M({\bf x}_m))_{m\in\mathbb{Z}_{\geqslant0}},
	\end{equation*}
	where ${\bf x}_0:=\emptyset$ and ${\bf x}_m:=\varia{x_1,\ \dots,\ x_m}{\sigma_1,\ \dots,\ \sigma_m}$ for $m\geqslant1$. Under the notations,  the product of $ A, B\in \mathrm{ARI}(\Gamma)$ is expressed as
	\begin{equation*}
		 A\times  B
	=\left(\sum_{\substack{
		{\bf x}_m=\alpha\beta \\

		}}A^{l(\alpha)}(\alpha)B^{l(\beta)}(\beta)\right)_{m\in\mathbb{Z}_{\geqslant0}}
	\end{equation*}
	where $\alpha$ and $\beta$ run over  $X_{\mathbb{Z}}^\bullet$.

We set
$\mathcal A_X:=\mathbb Q \langle X_{\mathbb{Z}} \rangle$
to be the non-commutative polynomial algebra generated by
$X_{\mathbb{Z}}$
(i.e. $\mathcal A_X$ is the $\mathbb Q$-linear space generated by $X_{\mathbb{Z}}^{\bullet}$).
\footnote{
We should beware of the inequality
$\varia{x_1+x_2}{\quad\sigma}\neq\varia{x_1}{\sigma}+\varia{x_2}{\sigma}$ and
$\varia{0}{\sigma}\neq0$.
}
We equip $\mathcal A_X$ a product $\scalebox{.8}{$\Sha$}:\mathcal A_X^{\otimes2} \rightarrow \mathcal A_X$
which is linearly defined by $\emptyset\ \scalebox{.8}{$\Sha$}\ \omega:=\omega\ \scalebox{.8}{$\Sha$}\ \emptyset:=w$ and
\begin{equation}
	u\omega\ \scalebox{.8}{$\Sha$}\ v\eta:=u(\omega\ \scalebox{.8}{$\Sha$}\ v\eta)+v(u\omega\ \scalebox{.8}{$\Sha$}\ \eta),
\end{equation}
for $u,v\in X_{\mathbb{Z}}$
and $\omega,\eta\in X_{\mathbb{Z}}^\bullet$.
Then the pair $(\mathcal A_X,\scalebox{.8}{$\Sha$})$ forms a commutative, associative, unital $\mathbb Q$-algebra.

We define the family $\left\{\Sh{\omega}{\eta}{\alpha}\right\}_{\omega,\eta,\alpha\in X_{\mathbb{Z}}^\bullet}$ in $\mathbb{Z}$ by
\begin{equation*}
	\omega\ \scalebox{.8}{$\Sha$}\ \eta
	=\sum_{\alpha\in X_{\mathbb{Z}}^\bullet}
	\Sh{\omega}{\eta}{\alpha}\alpha.
\end{equation*}
Particularly for $p,q\in\mathbb{N}$ and $u_1,\dots,u_{p+q}\in X_{\mathbb{Z}}$, we rewrite the shuffle product by
\begin{equation*}
	(u_1,\dots,u_p)\ \scalebox{.8}{$\Sha$}\ (u_{p+1},\dots,u_{p+q})=\sum_{\sigma\in\Sha_{p,q}}(u_{\sigma(1)},\dots,u_{\sigma(p)},u_{\sigma(p+1)},\dots,u_{\sigma(p+q)}).
\end{equation*}
Here the set $\Sha_{p,q}$ is defined by
\begin{equation}\label{shuffle permutation}
	\{\sigma\in S_{p+q}
	\ |\ \sigma^{-1}(1)<\cdots<\sigma^{-1}(p),\ \sigma^{-1}(p+1)<\cdots<\sigma^{-1}(p+q)\},
\end{equation}
where $S_{p+q}$ is the symmetry group with degree $p+q$.

\begin{defn}\label{def:ARIal}
A mould $M \in\mathrm{ARI}(\Gamma)$ is called {\it alternal} (cf. \cite[I--p.118]{E81}) if
\begin{equation}\label{eq:alternal}
\sum_{\alpha\in X_{\mathbb{Z}}^\bullet}\Sh{\varia{x_1,\ \dots,\ x_p}{\sigma_1,\ \dots,\ \sigma_p}}{\varia{x_{p+1},\ \dots,\ x_{p+q}}{\sigma_{p+1},\ \dots,\ \sigma_{p+q}}}{\alpha} M^{p+q}(\alpha)=0,
\end{equation}
for all $p,q\geqslant1$.
The $\mathbb Q$-linear space $\mathrm{ARI}(\Gamma)_\mathrm{al}$ is defined to be the subset of moulds $ M\in\mathrm{ARI}(\Gamma)$ which are alternal
(cf. \cite{E-ARIGARI}, \cite{E-flex}).
\end{defn}

We encode it
 with the induced depth filtrations and it forms a filtered $\mathbb Q$-algebra.

\begin{rem}\label{rem:component of mould by embedding}
Assume that $u_1,\dots,u_m\in \mathcal F$ are algebraically independent over $\mathbb Q$.
For $ M\in\mathrm{ARI}(\Gamma)$ we denote
$		 M^m\varia{u_1,\ \dots,\ u_m}{\sigma_1,\ \dots,\ \sigma_m}$
to be the image of $M^m\varia{x_1,\ \dots,\ x_m}{\sigma_1,\ \dots,\ \sigma_m}$ under the field embedding
$\mathbb Q(x_1,\dots,x_m)\hookrightarrow \mathcal F$
sending $x_i\mapsto u_i$.
\end{rem}
For our later use, we  prepare more notations:

\begin{nota}[{\cite[\S 2.1]{E-flex}}]\label{nota:mould operations}
For any mould $M=\left( M^m\varia{u_1,\ \dots,\ u_m}{\sigma_1,\ \dots,\ \sigma_m}\right)_m\in\mathrm{ARI}(\Gamma)$,  we define
\begin{align*}
&\mathrm{mantar}( M)^m
{\scriptsize\left(\begin{array}{rrr}
	u_1,& \dots,& u_m \\
	\sigma_1,& \dots,& \sigma_m
\end{array}\right)}
	=(-1)^{m-1} M^m
	{\scriptsize\left(\begin{array}{rrr}
		u_m,& \dots,& u_1 \\
		\sigma_m,& \dots,& \sigma_1
	\end{array}\right)},\\
&\mathrm{push}( M)^m
{\scriptsize\left(\begin{array}{rrr}
	u_1,& \dots,& u_m \\
	\sigma_1,& \dots,& \sigma_m
\end{array}\right)}
	= M^m
	{\scriptsize\left(\begin{array}{rrrr}
		-u_1-\cdots-u_m,& u_1,& \dots,& u_{m-1} \\
		\sigma_m^{-1},& \sigma_1\sigma_m^{-1},& \dots,& \sigma_{m-1}\sigma_m^{-1}
	\end{array}\right)},\\
&\mathrm{neg}( M)^m
{\scriptsize\left(\begin{array}{rrr}
	u_1,& \dots,& u_m \\
	\sigma_1,& \dots,& \sigma_m
\end{array}\right)}
	= M^m
	{\scriptsize\left(\begin{array}{rrr}
		-u_1,& \dots,& -u_m \\
		\sigma_1^{-1},& \dots,& \sigma_m^{-1}
\end{array}\right)},\\
&\mathrm{teru}( M)^m
{\scriptsize\left(\begin{array}{rrr}
	u_1,& \dots,& u_m \\
	\sigma_1,& \dots,& \sigma_m
\end{array}\right)}
	= M^m
	{\scriptsize\left(\begin{array}{rrr}
		u_1,& \dots,& u_m \\
		\sigma_1,& \dots,& \sigma_m
\end{array}\right)} \\
&\qquad+\frac{1}{u_m}\left\{ M^{m-1}
	{\scriptsize\left(\begin{array}{rrrl}
		u_1,& \dots,& u_{m-2},& u_{m-1}+u_m \\
		\sigma_1,& \dots,& \sigma_{m-2},& \sigma_{m-1}
	\end{array}\right)}
	- M^{m-1}
	{\scriptsize\left(\begin{array}{rrr}
		u_1,& \dots,& u_{m-1} \\
		\sigma_1,& \dots,& \sigma_{m-1}
	\end{array}\right)}\right\}.
\end{align*}
\end{nota}
Note that they are all $\mathbb Q$-linear endomorphisms on $\mathrm{ARI}(\Gamma)$.
We remark that $\mathrm{neg}\circ\mathrm{neg}=\mathrm{id}$ and
 $\mathrm{mantar}\circ\mathrm{mantar}=\mathrm{id}$.

\begin{defn}
We call a mould $M\in\mathrm{ARI}(\Gamma)$ {\it push-invariant}
when we have
\begin{equation}\label{push-invariant}
	\mathrm{push}( M)= M.
\end{equation}
We define $\mathrm{ARI}(\Gamma)_{\mathrm{push}}$ (\cite[\S 2.5]{E-flex})
to be the set of moulds $ M$ in $\mathrm{ARI}(\Gamma)$
which is push-invariant \eqref{push-invariant}.
\end{defn}

\subsection{Flexions and ari-bracket}\label{subsec:ari-bracket}
We explain an ari-bracket $\mathrm{ari}_u$ on $\mathrm{ARI}(\Gamma)$
by using flexions.

The notion of flexions  is introduced by Ecalle in \cite[\S 2.1]{E-flex}
for bimoulds (cf. \cite[\S 2.2]{S-ARIGARI}).
Here we consider those for moulds in $\mathrm{ARI}(\Gamma)$.
\begin{defn}\label{def:flexion}
The {\it flexions} are the
 four binary operators $\urflex{*}{*},\ \ulflex{*}{*},\ \lrflex{*}{*},\ \llflex{*}{*}:X_{\mathbb{Z}}^\bullet\times X_{\mathbb{Z}}^\bullet\rightarrow X_{\mathbb{Z}}^\bullet$
 which are defined by
\begin{align*}
	\urflex{\beta}{\alpha}
	&:= {\scriptsize\left(\begin{array}{rrrr}
		b_1+\cdots+b_n+a_1,& a_2,& \dots,& a_m \\
		\sigma_1,& \sigma_2,& \dots,& \sigma_m
	\end{array}\right)} , \\
	\ulflex{\alpha}{\beta}
	&:= {\scriptsize\left(\begin{array}{rrrl}
		a_1,& \dots,& a_{m-1},& a_m+b_1+\cdots+b_n \\
		\sigma_1,& \dots,& \sigma_{m-1},& \sigma_m
	\end{array}\right)} , \\
	\lrflex{\beta}{\alpha}
	&:={\scriptsize\left(\begin{array}{rrr}
		a_1,& \dots,& a_m \\
		\tau_n^{-1}\sigma_1,& \dots,& \tau_n^{-1}\sigma_m
	\end{array}\right)} , \\
	\llflex{\alpha}{\beta}
	&:={\scriptsize\left(\begin{array}{rrr}
		a_1,& \dots,& a_m \\
		\sigma_1\tau_1^{-1},& \dots,& \sigma_m\tau_1^{-1}
	\end{array}\right)}  , \\
	\urflex{\emptyset}{\gamma}&:=\ulflex{\gamma}{\emptyset}:=\lrflex{\emptyset}{\gamma}:=\llflex{\gamma}{\emptyset}:=\gamma , \\
	\urflex{\gamma}{\emptyset}&:=\ulflex{\emptyset}{\gamma}:=\lrflex{\gamma}{\emptyset}:=\llflex{\emptyset}{\gamma}:=\emptyset ,
\end{align*}
for $\alpha=\varia{a_1,\dots,a_m}{\sigma_1,\dots,\sigma_m},\beta=\varia{b_1,\dots,b_n}{\tau_1,\dots,\tau_n}\in X_{\mathbb{Z}}^\bullet$ ($m,n\geqslant1$) and $\gamma\in X_{\mathbb{Z}}^\bullet$.
\end{defn}
	Note that we have

		$
		l(\urflex{\beta}{\alpha})=l(\ulflex{\alpha}{\beta})=l(\lrflex{\beta}{\alpha})=l(\llflex{\alpha}{\beta})=l(\alpha) \text{ and }

		l(\alpha,\beta)=l(\alpha)+l(\beta)
		$
	for $\alpha,\beta\in X_{\mathbb{Z}}^\bullet$.

The derivation $\mathrm{arit}$ and bracket $\mathrm{ari}$  are introduced for bimoulds in terms of flexions in \cite[\S 2.2]{E-flex} (cf. \cite[\S 2.2]{S-ARIGARI}) and
here we consider those for $\mathrm{ARI}(\Gamma)$ as follows.

\begin{defn}\label{def:aritu}
      Let $ B\in \mathcal \mathrm{ARI}(\Gamma)$. The linear map
      \footnote{The lower suffix $u$ is reflected by the notion of $u$-moulds in \cite{S-ARIGARI}.}
     $\mathrm{arit}_u( B):\mathrm{ARI}(\Gamma) \rightarrow \mathrm{ARI}(\Gamma)$ is defined by
     \footnote{\label{footnote:arit embedding} For $m\geqslant2$ and $\alpha,\beta,\gamma\in X_\mathbb{Z}^\bullet$ with ${\bf x}_m=\alpha\beta\gamma$, all letters appearing in the word $\alpha\urflex{\beta}{\gamma}$ (resp. $\lrflex{\beta}{\gamma}$) are algebraically independent over $\mathbb Q$. So by Remark \ref{rem:component of mould by embedding}, the component $ A^{l(\alpha,\gamma)}(\alpha\urflex{\beta}{\gamma})$ (resp. $B^{l(\beta)}(\llflex{\beta}{\gamma})$) is well-defined. Similarly, $ A^{l(\alpha,\gamma)}(\ulflex{\alpha}{\beta}\gamma)$ and $B^{l(\beta)}(\lrflex{\alpha}{\beta})$ are also well-defined.}
	\begin{align*}
		&(\mathrm{arit}_u( B)( A))^m({\bf x}_m)
		=(\mathrm{arit}_u( B)( A))^m\varia{x_1,\ \dots,\ x_m}{\sigma_1,\ \dots,\ \sigma_m} \\
		&:=\left\{\begin{array}{ll}
			\displaystyle \sum_{\substack{{\bf x}_m=\alpha\beta\gamma \\
				\beta,\gamma\neq\emptyset}}
			 A^{l(\alpha,\gamma)}(\alpha\urflex{\beta}{\gamma}) B^{l(\beta)}(\llflex{\beta}{\gamma})
			-\sum_{\substack{{\bf x}_m=\alpha\beta\gamma \\
				\alpha,\beta\neq\emptyset}}
			 A^{l(\alpha,\gamma)}(\ulflex{\alpha}{\beta}\gamma) B^{l(\beta)}(\lrflex{\alpha}{\beta}) & (m\geqslant2),\\
			0 & (m=0,1),
		\end{array}\right.
	\end{align*}
	for $A\in\mathrm{ARI}(\Gamma)$.
\end{defn}

It is shown in \cite[Appendix A.1]{S-ARIGARI}  that
$\mathrm{arit}$ forms a derivation on the set of bimoulds.
The same holds for  $\mathrm{ARI}(\Gamma)$  as follows,
where we present an alternative  proof  to hers.

\begin{lem}\label{lem:arit derivation}
For any $A\in\mathrm{ARI}(\Gamma)$, $\mathrm{arit}_u(A)$ forms a derivation of $\mathrm{ARI}(\Gamma)$
with respect to the product $\times$,
 that is, for any $B,C\in\mathrm{ARI}(\Gamma)$, we have
\begin{equation*}
	\mathrm{arit}_u(A)(B\times C)=\mathrm{arit}_u(A)(B)\times C+B\times \mathrm{arit}_u(A)(C).
\end{equation*}
\end{lem}

\begin{proof}
Let $m\geqslant0$. We have
{\footnotesize
\begin{align*}
	(\mathrm{arit}(A) & (B \times C))({\bf x}_m)
	= \sum_{\substack{
		{\bf x}_m=abc \\
		b,c\neq\emptyset}}
	(B\times C)(a \urflex{b}{c})A(\llflex{b}{c})
	-\sum_{\substack{
		{\bf x}_m=abc \\
		a,b\neq\emptyset}}
	(B\times C)(\ulflex{a}{b} c)A(\lrflex{a}{b}) \\
	=& \sum_{\substack{
		{\bf x}_m=abc \\
		b,c\neq\emptyset}}
	\left\{
		\sum_{a \urflex{b}{c}=de}
		B(d)C(e)
	\right\}A(\llflex{b}{c})
	-\sum_{\substack{
		{\bf x}_m=abc \\
		a,b\neq\emptyset}}
	\left\{
		\sum_{\ulflex{a}{b}c=ef}
		B(e)C(f)
	\right\}A(\lrflex{a}{b})
	\intertext{{\normalsize
By applying Lemma \ref{lem:tri-factorization}.(1) for $c\neq\emptyset$, $f=\emptyset$ to the first term and by applying Lemma \ref{lem:tri-factorization}.(2) for $a\neq\emptyset$, $d=\emptyset$ to the second term, we have}}
	=& \sum_{\substack{
		{\bf x}_m=abc \\
		b,c\neq\emptyset}}
	\left\{
	\sum_{a=a_1a_2}
	B(a_1)C(a_2 \urflex{b}{c})
	+\sum_{\substack{
		c=c_1c_2 \\
		c_1\neq\emptyset}}
	B(a \urflex{b}{c_1})C(c_2)
	\right\}A(\llflex{b}{c}) \\
	&-\sum_{\substack{
		{\bf x}_m=abc \\
		a,b\neq\emptyset}}
	\left\{
	\sum_{\substack{
		a=a_1a_2 \\
		a_2\neq\emptyset}}
	B(a_1)C(\ulflex{a_2}{b} c)
	+\sum_{c=c_1c_2}
	B(\ulflex{a}{b} c_1)C(c_2)
	\right\}A(\lrflex{a}{b}) \\
	=& \sum_{\substack{
		{\bf x}_m=a_1a_2bc \\
		b,c\neq\emptyset}}
	B(a_1)C(a_2 \urflex{b}{c})A(\llflex{b}{c})
	+\sum_{\substack{
		{\bf x}_m=abc_1c_2 \\
		b,c_1\neq\emptyset}}
	B(a \urflex{b}{c_1})C(c_2)A(\llflex{b}{c_1c_2}) \\
	&-\sum_{\substack{
		{\bf x}_m=a_1a_2bc \\
		a_2,b\neq\emptyset}}
	B(a_1)C(\ulflex{a_2}{b}c)A(\lrflex{a_1a_2}{b})
	-\sum_{\substack{
		{\bf x}_m=abc_1c_2 \\
		a,b\neq\emptyset}}
	B(\ulflex{a}{b} c_1)C(c_2)A(\lrflex{a}{b}).
	\intertext{{\normalsize
By applying Lemma \ref{lem:flexion's action} to the second term, we get $A(\llflex{b}{c_1c_2})=A(\llflex{b}{c_1})$. Similarly, for the third term, we get $A(\lrflex{a_1a_2}{b})=A(\lrflex{a_2}{b})$. Therefore, we calculate}}
	=& \sum_{{\bf x}_m=dc_2}
	\left\{
	\sum_{\substack{
		d=abc_1 \\
		b,c_1\neq\emptyset}}
	B(a \urflex{b}{c_1})A(\llflex{b}{c_1})
	-\sum_{\substack{
		d=abc_1 \\
		a,b\neq\emptyset}}
	B(\ulflex{a}{b} c_1)A(\lrflex{a}{b})
	\right\}C(c_2) \\
	&+\sum_{{\bf x}_m=a_1d}
	B(a_1)
	\left\{
	\sum_{\substack{
		d=a_2bc \\
		b,c\neq\emptyset}}
	C(a_2 \urflex{b}{c})A(\llflex{b}{c})
	-\sum_{\substack{
		d=a_2bc \\
		a_2,b\neq\emptyset}}
	C(\ulflex{a_2}{b} c)A(\lrflex{a_2}{b})
	\right\} \\
	=& \sum_{{\bf x}_m=dc_2}
	(\mathrm{arit}(A)(B))(d)C(c_2)
	+\sum_{\bf w_n=a_1d}
	B(a_1)(\mathrm{arit}(A)(C))(d) \\
	=& (\mathrm{arit}(A)(B)\times C)({\bf x}_m)
	+(B\times\mathrm{arit}(A)(C))({\bf x}_m).
\end{align*}
}
Hence, we obtain the claim.
\end{proof}

We define the following bracket as with the bracket $\mathrm{ari}$ introduced in \cite[(2.40)]{E-flex}.
\begin{defn}
The {\it $\mathrm{ari}_u$-bracket} means
the bilinear map $\mathrm{ari}_u:\mathrm{ARI}(\Gamma)^{\otimes2}\rightarrow \mathrm{ARI}(\Gamma)$ which is defined by
\begin{equation}\label{ari-bracket}
	\mathrm{ari}_u( A, B)
	:=\mathrm{arit}_u( B)( A)-\mathrm{arit}_u( A)( B)
	+[ A,  B]
\end{equation}
for $ A, B\in\mathrm{ARI}(\Gamma)$.
Here
\footnote{In the papers \cite{E-flex}, \cite{S-ARIGARI} and \cite{RS}, the product $A\times B$ (resp. the bracket $[A,B]$) is denoted by $mu(A,B)$ (resp. $lu(A,B)$).}
, we have

$ [A, B]:= A\times  B- B\times  A$.
\end{defn}

We note that the bracket $\mathrm{ari}_u(A,  B)$  in the case when $\Gamma=\{e\}$
also appears in \cite[(A.3)]{R-PhD}
and is denoted by  $[ A, B]_{\mathrm{ari}}$.

The following is also stated for moulds and bimoulds
in \cite[Proposition 2.2.2]{S-ARIGARI}
where her key formula (2.2.10) looks unproven and containing a signature error.

\begin{prop}\label{ARI Lie algebra}
	The $\mathbb Q$-linear space $\mathrm{ARI}(\Gamma)$ forms a filtered Lie algebra under the $\mathrm{ari}_u$-bracket.
\end{prop}
\begin{proof}
	We give a self-contained proof in Appendix \ref{sec:A.1}.
\end{proof}

The following proposition for $\Gamma=\{e\}$ is shown in \cite[Appendix A]{SS}.
\begin{prop
\label{ARIal Lie algebra}
The $\mathbb Q$-linear space $\mathrm{ARI}(\Gamma)_\mathrm{al}$ forms a filtered Lie subalgebra of $\mathrm{ARI}(\Gamma)$ under the $\mathrm{ari}_u$-bracket.
\end{prop}
\begin{proof}
	We prove this in Appendix \ref{sec:A.2}.
\end{proof}

\subsection{Swap and bialternality}\label{subsec:swap}
We encode  $\mathrm{ARI}(\Gamma)$ with another Lie algebra structure introduced by $\mathrm{ari}_v$.
We prepare $\overline{\mathrm{ARI}}(\Gamma)$, a copy of $\mathrm{ARI}(\Gamma)$.
We denote
$M^m_{\sigma_1, \dots, \sigma_m}(x_1,\ \dots,\ x_m)$ by $M^m
\varia{\sigma_1,\ \dots,\ \sigma_m}{x_1,\ \dots,\ x_m}$
\footnote{
We note that the top and bottom rows are switched.
}
for each element $M$ in $\overline{\mathrm{ARI}}(\Gamma)$
to distinguish it from an element in  $\mathrm{ARI}(\Gamma)$.

Similarly to our previous sections, we work over the following algebraic formulation:
Put $Y:=\left\{\binom{\sigma}{y_i}\right\}_{i\in\mathbb{N},\sigma\in\Gamma}$. Let $Y_{\mathbb{Z}}$ be the set such that
$$Y_{\mathbb{Z}}:=\left\{\varia{\sigma}{v}\ \middle|\ v=a_1y_1+\cdots +a_ky_k,\ k\in\mathbb{N},\ a_j\in\mathbb{Z},\ \sigma\in\Gamma\right\},$$
and let
$Y_{\mathbb{Z}}^\bullet$
be the non-commutative free monoid generated by all elements of $Y_{\mathbb{Z}}$
with the empty word $\emptyset$ as the unit.
We set
$\mathcal A_Y:=\mathbb Q \langle Y_{\mathbb{Z}} \rangle$
to be the non-commutative polynomial algebra generated by
$Y_{\mathbb{Z}}$.
In the same way to $\mathcal A_X$,
it is equipped with a structure of
a commutative, associative, unital $\mathbb Q$-algebra
with
the shuffle product $\scalebox{.8}{$\Sha$}:\mathcal A_Y^{\otimes2} \rightarrow \mathcal A_Y$ .
The flexions are also introduced in this setting.
\begin{defn}
The {\it flexions} on  $Y_{\mathbb{Z}}^\bullet$ are the
 four binary operators $\urflex{*}{*},\ \ulflex{*}{*},\ \lrflex{*}{*},\ \llflex{*}{*}:Y_{\mathbb{Z}}^\bullet\times Y_{\mathbb{Z}}^\bullet\rightarrow Y_{\mathbb{Z}}^\bullet$
 which are defined by
\begin{align*}
	\urflex{\beta}{\alpha}
	&:= {\scriptsize\left(\begin{array}{crrr}
		\sigma_1\tau_1\cdots\tau_n,& \sigma_2,& \dots,& \sigma_m \\
		a_1,& a_2,& \dots,& a_m
	\end{array}\right)} , \\
	\ulflex{\alpha}{\beta}
	&:= {\scriptsize\left(\begin{array}{rrrc}
		\sigma_1,& \dots,& \sigma_{m-1},& \sigma_m\tau_1\cdots\tau_n \\
		a_1,& \dots,& a_{m-1},& a_m
	\end{array}\right)} , \\
	\lrflex{\beta}{\alpha}
	&:={\scriptsize\left(\begin{array}{crc}
		\sigma_1,& \dots,& \sigma_m \\
		a_1-b_n,& \dots,& a_m-b_n
	\end{array}\right)} , \\
	\llflex{\alpha}{\beta}
	&:={\scriptsize\left(\begin{array}{crc}
		\sigma_1,& \dots,& \sigma_m \\
		a_1-b_1,& \dots,& a_m-b_1
	\end{array}\right)}  , \\
	\urflex{\emptyset}{\gamma}&:=\ulflex{\gamma}{\emptyset}:=\lrflex{\emptyset}{\gamma}:=\llflex{\gamma}{\emptyset}:=\gamma , \\
	\urflex{\gamma}{\emptyset}&:=\ulflex{\emptyset}{\gamma}:=\lrflex{\gamma}{\emptyset}:=\llflex{\emptyset}{\gamma}:=\emptyset ,
\end{align*}
for $\alpha=\varia{\sigma_1,\dots,\sigma_m}{a_1,\dots,a_m},\beta=\varia{\tau_1,\dots,\tau_n}{b_1,\dots,b_n}\in Y_{\mathbb{Z}}^\bullet$ ($m,n\geqslant1$) and $\gamma\in Y_{\mathbb{Z}}^\bullet$.
\end{defn}

We denote ${\bf y}_0:=\emptyset$ and ${\bf y}_m:=\varia{\sigma_1,\ \dots,\ \sigma_m}{y_1,\ \dots,\ y_m}$ for $m\geqslant1$.
\begin{defn
      Let $ B\in \overline{\mathrm{ARI}}(\Gamma)$. The linear map $\mathrm{arit}_v( B):\overline{\mathrm{ARI}}(\Gamma) \rightarrow \overline{\mathrm{ARI}}(\Gamma)$ is defined by
            \footnote{The lower suffix $v$ is reflected by the notion of $v$-moulds in \cite{S-ARIGARI}.}
	\begin{align*}
		&(\mathrm{arit}_v( B)( A))^m({\bf y}_m)
		=(\mathrm{arit}_v( B)( A))^m\varia{\sigma_1,\ \dots,\ \sigma_m}{y_1,\ \dots,\ y_m} \\
		&:=\left\{\begin{array}{ll}
			\displaystyle \sum_{\substack{{\bf y}_m=\alpha\beta\gamma \\
				\beta,\gamma\neq\emptyset}}
			 A^{l(\alpha,\gamma)}(\alpha\urflex{\beta}{\gamma}) B^{l(\beta)}(\llflex{\beta}{\gamma})
			-\sum_{\substack{{\bf y}_m=\alpha\beta\gamma \\
				\alpha,\beta\neq\emptyset}}
			 A^{l(\alpha,\gamma)}(\ulflex{\alpha}{\beta}\gamma) B^{l(\beta)}(\lrflex{\alpha}{\beta}) & (m\geqslant2),\\
			0 & (m=0,1),
		\end{array}\right.
	\end{align*}
	for $A\in\overline{\mathrm{ARI}}(\Gamma)$.
\end{defn}
Similarly to Lemma \ref{lem:arit derivation}, the following holds.
\begin{lem}\label{lem:aritv derivation}
For any $A\in\overline{\mathrm{ARI}}(\Gamma)$, $\mathrm{arit}_v(A)$ forms a derivation of $\overline{\mathrm{ARI}}(\Gamma)$
with respect to the product $\times$.
\end{lem}
\begin{proof}
The $\mathrm{arit}_v$-bracket can be expressed by the exactly same formula as the $\mathrm{arit}_u$-bracket in terms of flexions.
Therefore it can be proved in the same way to the one of
Lemma \ref{lem:arit derivation}.
\end{proof}

\begin{defn}
The {\it $\mathrm{ari}_v$-bracket} means
the bilinear map $\mathrm{ari}_v:\overline{\mathrm{ARI}}(\Gamma)^{\otimes2}\rightarrow \overline{\mathrm{ARI}}(\Gamma)$ which is defined by
\footnote{In \cite{RS}, the brackets $\mathrm{arit}_v( A, B)$ and $\mathrm{ari}_v( A, B)$ are denoted by $\overline{\mathrm{arit}}$ and $\overline{\mathrm{ari}}$ respectively.}
\begin{equation}\label{ari v-bracket}
	\mathrm{ari}_v( A, B)
	:=\mathrm{arit}_v( B)( A)-\mathrm{arit}_v( A)( B)
	+[ A,  B]
\end{equation}
for $ A, B\in\overline{\mathrm{ARI}}(\Gamma)$.
Here
$ [A, B]:= A\times  B- B\times  A$.
\end{defn}
Similarly to Proposition \ref{ARI Lie algebra}, the following holds.
\begin{prop}\label{barARI Lie algebra}
	The $\mathbb Q$-linear space $\overline{\mathrm{ARI}}(\Gamma)$ forms a filtered Lie algebra under the $\mathrm{ari}_v$-bracket.
\end{prop}
\begin{proof}
It can be also proved in the same way to the one of
Proposition \ref{ARI Lie algebra}.
\end{proof}

In the same way to Definition \ref{def:ARIal},
alternal moulds in $\overline{\mathrm{ARI}}(\Gamma)$ can be introduced and we denote  $\overline{\mathrm{ARI}}(\Gamma)_\mathrm{al}$ to be
its subset consisting of  alternal moulds.
Similarly to Proposition \ref{ARIal Lie algebra}, the following holds.

\begin{prop}\label{barARIal Lie algebra}
$\overline{\mathrm{ARI}}(\Gamma)_\mathrm{al}$ forms a Lie algebra under the $\mathrm{ari}_v$-bracket.
\end{prop}
\begin{proof}
It can be proved in the same way to the one of
Proposition \ref{ARIal Lie algebra}.
\end{proof}

We define the $\mathbb Q$-linear map $\mathrm{swap}:\mathrm{ARI}(\Gamma) \rightarrow \overline{\mathrm{ARI}}(\Gamma)$ by
\begin{equation}\label{eq:swap}
	\mathrm{swap}( M)^m
{\scriptsize\left(\begin{array}{rrr}
	\sigma_1,& \dots,& \sigma_m \\
	v_1,& \dots,& v_m
\end{array}\right)}
= M^m
{\scriptsize\left(\begin{array}{rrrrr}
	v_m,& v_{m-1}-v_m,& \dots,& v_2-v_3,& v_1-v_2 \\
	\sigma_1\cdots\sigma_m,& \sigma_1\cdots\sigma_{m-1},& \dots,& \sigma_1\sigma_2,& \sigma_1
\end{array}\right)}
\end{equation}
for any mould $M=\left(M^m\varia{u_1,\ \dots,\ u_m}{\sigma_1,\ \dots,\ \sigma_m}\right)\in\mathrm{ARI}(\Gamma)$.

\begin{defn}[cf. \cite{E-ARIGARI}, \cite{E-flex}]\label{def:ARIalal}
The subset
$\mathrm{ARI}(\Gamma)_{\underline\mathrm{al}/\underline\mathrm{al}}$
of bialternal moulds
is defined to be
\begin{align*}
	\mathrm{ARI}(\Gamma)_{\underline\mathrm{al}/\underline\mathrm{al}}:=&\{M\in\mathrm{ARI}(\Gamma)_\mathrm{al} \ |\
	\mathrm{swap}(M)\in\overline{\mathrm{ARI}}(\Gamma)_\mathrm{al}, \  M^1\varia{x_1}{\sigma_1}=M^1\varia{-x_1}{\sigma_1^{-1}}\}

\end{align*}
\end{defn}

\begin{prop
\label{ARIalal Lie algebra}
The $\mathbb Q$-linear space $\mathrm{ARI}(\Gamma)_{\underline\mathrm{al}/\underline\mathrm{al}}$
 forms a filtered Lie subalgebra of $\mathrm{ARI}(\Gamma)_\mathrm{al}$ under the $\mathrm{ari}_u$-bracket.
\end{prop}
\begin{proof}
	We prove this in Appendix \ref{sec:A.3}.
\end{proof}

\subsection{Push-invariance and pus-neutrality}\label{subsec:push-invariance and pus-neutrality}
We introduce the set  $\mathrm{ARI}(\Gamma)_{\mathrm{push}/\mathrm{pusnu}}$ of moulds which are push-invariant
and whose $\mathrm{swap}$ are pus-neutral and
show that it forms a Lie algebra under the ari-bracket.

By abuse of notation, we introduce with the following notations for $\overline{\mathrm{ARI}}(\Gamma)$
similarly to Notation \ref{nota:mould operations}.
\begin{nota}[{\cite[\S 2.1]{E-flex}}]
For any mould $M=\left( M^m\varia{\sigma_1,\ \dots,\ \sigma_m}{v_1,\ \dots,\ v_m}\right)_m\in\overline{\mathrm{ARI}}(\Gamma)$,  we define
\begin{align*}
&\mathrm{mantar}( M)^m
{\scriptsize\left(\begin{array}{rrr}
	\sigma_1,& \dots,& \sigma_m \\
	v_1,& \dots,& v_m
\end{array}\right)}
	=(-1)^{m-1} M^m
	{\scriptsize\left(\begin{array}{rrr}
		\sigma_m,& \dots,& \sigma_1 \\
		v_m,& \dots,& v_1
	\end{array}\right)},\\
&\mathrm{pus}( M)^m
{\scriptsize\left(\begin{array}{rrr}
	\sigma_1,& \dots,& \sigma_m \\
	v_1,& \dots,& v_m
\end{array}\right)}
	= M^m
	{\scriptsize\left(\begin{array}{rrrr}
		\sigma_m,& \sigma_1,& \dots,& \sigma_{m-1} \\
		v_m,& v_1,& \dots,& v_{m-1}
\end{array}\right)},\\
&\mathrm{neg}( M)^m
{\scriptsize\left(\begin{array}{rrr}
	\sigma_1,& \dots,& \sigma_m \\
	v_1,& \dots,& v_m
\end{array}\right)}
	= M^m
	{\scriptsize\left(\begin{array}{rrr}
		\sigma_1^{-1},& \dots,& \sigma_m^{-1} \\
		-u_1,& \dots,& -u_m
\end{array}\right)}.
\end{align*}
\end{nota}

\begin{defn
\label{def:ARIpushpusnu}
We call a mould $N\in\overline{\mathrm{ARI}}(\Gamma)$ {\it pus-neutral} (\cite[(2.73)]{E-flex})
\footnote{
It is not push-neutral but pus-neutral.
In \cite[Definition 5']{RS},  it is called {\it circ-neutral} when \eqref{pus-neutral} holds for $m>1$ when $\Gamma=\{e\}$.
}
when we have
\begin{equation}\label{pus-neutral}
	\sum_{i=1}^{m}\mathrm{pus}^i(N)^m\varia{\sigma_1,\ \dots,\ \sigma_m}{v_1,\ \dots,\ v_m}
	\left(=\sum_{i\in\mathbb{Z}/m\mathbb{Z}} N^m\varia{\sigma_{i+1},\ \dots,\ \sigma_{i+m}}{v_{i+1},\ \dots,\ v_{i+m}}\right)=0
\end{equation}
for all $m\geqslant1$ and $\sigma_1,\dots,\sigma_m\in\Gamma$.
We define  $\mathrm{ARI}(\Gamma)_{\mathrm{push}/ \mathrm{pusnu}}$
\footnote{
We expect that our $\mathrm{ARI}(\Gamma)_{\mathrm{push}/ \mathrm{pusnu}}$
might be related to Ecalle's $\mathrm{ARI}^{\ast/\overline{\mathrm{pusnu}}}$
(\cite[\S 2.5]{E-flex}),
whose precise definition looks missing.
}
to be the set  of
moulds $M\in \mathrm{ARI}(\Gamma)$  which is push-invariant \eqref{push-invariant} and
whose $\mathrm{swap} (M)$ is pus-neutral  \eqref{pus-neutral}.
\end{defn}

Firstly we show that the set $\mathrm{ARI}_{\mathrm{push}}$ forms a Lie algebra under the $\mathrm{ari}_u$-bracket
which was stated in \cite[\S 2.5]{E-flex} without a detailed proof.

\begin{prop}\label{prop:ARIpush Lie}
	If $ A, B\in\mathrm{ARI}(\Gamma)$ are $\mathrm{push}$-invariant, then $\mathrm{ari}_u( A, B)$ is $\mathrm{push}$-invariant.
\end{prop}
\begin{proof}

	Let $m\geqslant1$. We put $\omega=\varia{u_1,\ \dots,\ u_m}{\sigma_1,\ \dots,\ \sigma_m}$. We have
	\begin{align*}
		\mathrm{push}&( A\times B)(\omega) \\
		=&( A\times B){\scriptsize\left(\begin{array}{rrrr}
			-u_1-\cdots-u_m,& u_1,& \dots,& u_{m-1} \\
			\sigma_m^{-1},&\sigma_1\sigma_m^{-1},& \dots,& \sigma_{m-1}\sigma_m^{-1}
		\end{array}\right)} \\
		=& A(\emptyset) B{\scriptsize\left(\begin{array}{rrrr}
			-u_1-\cdots-u_m,& u_1,& \dots,& u_{m-1} \\
			\sigma_m^{-1},&\sigma_1\sigma_m^{-1},& \dots,& \sigma_{m-1}\sigma_m^{-1}
		\end{array}\right)} \\
		&+\sum_{i=1}^m A{\scriptsize\left(\begin{array}{rrrr}
			-u_1-\cdots-u_m,& u_1,& \dots,& u_{i-1} \\
			\sigma_m^{-1},&\sigma_1\sigma_m^{-1},& \dots,& \sigma_{i-1}\sigma_m^{-1}
		\end{array}\right)}
		 B{\scriptsize\left(\begin{array}{rrr}
			u_i,& \dots,& u_{m-1} \\
			\sigma_i\sigma_m^{-1},& \dots,& \sigma_{m-1}\sigma_m^{-1}
		\end{array}\right)}.
	\end{align*}
	Because $ A,  B$ are $\mathrm{push}$-invariant, we get
	\begin{align}\label{push on product}
		\mathrm{push}&( A\times B)(\omega) \\
		=& A(\emptyset) B(\omega)
		+ A(\omega) B(\emptyset) \nonumber \\
		&+\sum_{i=1}^{m-1}
		A{\scriptsize\left(\begin{array}{rrrr}
			u_1,& \dots,& u_{i-1},& u_i+\cdots+u_m \\
			\sigma_1,& \dots,& \sigma_{i-1},& \sigma_m
		\end{array}\right)}
		B{\scriptsize\left(\begin{array}{rrr}
			u_i,& \dots,& u_{m-1} \\
			\sigma_i\sigma_m^{-1},& \dots,& \sigma_{m-1}\sigma_m^{-1}
		\end{array}\right)}. \nonumber
	\end{align}
	On the other hand, for $\eta=\varia{u_0,\ u_1,\ \dots,\ u_{m-1}}{\tau_0,\ \tau_1,\ \dots,\ \tau_{m-1}}$, we have
	\begin{align*}
		&\mathrm{arit}_u( B)( A)(\eta) \\
		=&\sum_{\substack{\eta=\alpha\beta\gamma \\
			\beta,\gamma\neq\emptyset}}
		 A(\alpha\urflex{\beta}{\gamma}) B(\llflex{\beta}{\gamma})
		-\sum_{\substack{\eta=\alpha\beta\gamma \\
			\alpha,\beta\neq\emptyset}}
		 A(\ulflex{\alpha}{\beta}\gamma) B(\lrflex{\alpha}{\beta}) \\
		=&\sum_{0\leqslant i\leqslant j<m-1}
			 A{\scriptsize\left(\begin{array}{rrrcrrr}
			 	u_0,& \dots,& u_{i-1},& u_i+\cdots+u_{j+1},& u_{j+2},& \dots,& u_{m-1} \\
			 	\tau_0,& \dots,& \tau_{i-1},& \tau_{j+1},& \tau_{j+2},& \dots,& \tau_{m-1}
			\end{array}\right)}
			B{\scriptsize\left(\begin{array}{rrr}
				u_i,& \dots,& u_j \\
				\tau_i\tau_{j+1}^{-1},& \dots,& \tau_j\tau_{j+1}^{-1} \\
			\end{array}\right)} \\
		&-\sum_{1\leqslant i\leqslant j\leqslant m-1}
			 A{\scriptsize\left(\begin{array}{rrrcrrr}
			 	u_0,& \dots,& u_{i-2},& u_{i-1}+\cdots+u_{j},& u_{j+1},& \dots,& u_{m-1} \\
			 	\tau_0,& \dots,& \tau_{i-2},& \tau_{i-1},& \tau_{j+1},& \dots,& \tau_{m-1}
			\end{array}\right)}
			 B{\scriptsize\left(\begin{array}{rrr}
				u_i,& \dots,& u_j \\
				\tau_i\tau_{i-1}^{-1},& \dots,& \tau_j\tau_{i-1}^{-1} \\
			\end{array}\right)} \\
		=&\sum_{0\leqslant j<m-1}
			 A{\scriptsize\left(\begin{array}{rrrr}
			 	u_0+\cdots+u_{j+1},& u_{j+2},& \dots,& u_{m-1} \\
			 	\tau_{j+1},& \tau_{j+2},& \dots,& \tau_{m-1}
			\end{array}\right)}
			B{\scriptsize\left(\begin{array}{rrr}
				u_0,& \dots,& u_j \\
				\tau_0\tau_{j+1}^{-1},& \dots,& \tau_j\tau_{j+1}^{-1} \\
			\end{array}\right)} \\
		&-\sum_{1\leqslant j\leqslant m-1}
			 A{\scriptsize\left(\begin{array}{rrrr}
			 	u_0+\cdots+u_{j},& u_{j+1},& \dots,& u_{m-1} \\
			 	\tau_{0},& \tau_{j+1},& \dots,& \tau_{m-1}
			\end{array}\right)}
			 B{\scriptsize\left(\begin{array}{rrr}
				u_1,& \dots,& u_j \\
				\tau_1\tau_{0}^{-1},& \dots,& \tau_j\tau_{0}^{-1} \\
			\end{array}\right)} \\
		&+\sum_{0< i\leqslant j<m-1}
			 A{\scriptsize\left(\begin{array}{rrrcrrr}
			 	u_0,& \dots,& u_{i-1},& u_i+\cdots+u_{j+1},& u_{j+2},& \dots,& u_{m-1} \\
			 	\tau_0,& \dots,& \tau_{i-1},& \tau_{j+1},& \tau_{j+2},& \dots,& \tau_{m-1}
			\end{array}\right)}
			B{\scriptsize\left(\begin{array}{rrr}
				u_i,& \dots,& u_j \\
				\tau_i\tau_{j+1}^{-1},& \dots,& \tau_j\tau_{j+1}^{-1} \\
			\end{array}\right)} \\
		&-\sum_{1< i\leqslant j\leqslant m-1}
			 A{\scriptsize\left(\begin{array}{rrrcrrr}
			 	u_0,& \dots,& u_{i-2},& u_{i-1}+\cdots+u_{j},& u_{j+1},& \dots,& u_{m-1} \\
			 	\tau_0,& \dots,& \tau_{i-2},& \tau_{i-1},& \tau_{j+1},& \dots,& \tau_{m-1}
			\end{array}\right)}
			 B{\scriptsize\left(\begin{array}{rrr}
				u_i,& \dots,& u_j \\
				\tau_i\tau_{i-1}^{-1},& \dots,& \tau_j\tau_{i-1}^{-1} \\
			\end{array}\right)}.
	\end{align*}
	So by putting $u_0=-u_1-\cdots-u_m$ and $\tau_0=\sigma_m^{-1}$ and $\tau_i=\sigma_i\sigma_m^{-1}$ for $1\leqslant i\leqslant m-1$, we calculate
	\begin{align*}
		&\mathrm{push}(\mathrm{arit}_u( B)( A))(\omega) \\
		=&(\mathrm{arit}_u( B)( A)){\scriptsize\left(\begin{array}{rrrr}
			-u_1-\cdots-u_m,& u_1,& \dots,& u_{m-1} \\
			\sigma_m^{-1},&\sigma_1\sigma_m^{-1},& \dots,& \sigma_{m-1}\sigma_m^{-1}
		\end{array}\right)} \\
		=&\sum_{0\leqslant j<m-1}
			 A{\scriptsize\left(\begin{array}{rrrr}
			 	-u_{j+2}-\cdots-u_m,& u_{j+2},& \dots,& u_{m-1} \\
			 	\sigma_{j+1}\sigma_m^{-1},& \sigma_{j+2}\sigma_m^{-1},& \dots,& \sigma_{m-1}\sigma_m^{-1}
			\end{array}\right)} \\
			&\hspace{5cm}\cdot B{\scriptsize\left(\begin{array}{rrrr}
				-u_1-\cdots-u_m,& u_1,& \dots,& u_j \\
				\sigma_{j+1}^{-1},& \sigma_1\sigma_{j+1}^{-1},& \dots,& \sigma_j\sigma_{j+1}^{-1} \\
			\end{array}\right)} \\
		&-\sum_{1\leqslant j\leqslant m-1}
			 A{\scriptsize\left(\begin{array}{rrrr}
			 	-u_{j+1}-\cdots-u_m,& u_{j+1},& \dots,& u_{m-1} \\
			 	\sigma_m^{-1},& \sigma_{j+1}\sigma_m^{-1},& \dots,& \sigma_{m-1}\sigma_m^{-1}
			\end{array}\right)} \\
			&\hspace{5cm}\cdot B{\scriptsize\left(\begin{array}{rrr}
				u_1,& \dots,& u_j \\
				\sigma_1,& \dots,& \sigma_j \\
			\end{array}\right)} \\
		&+\sum_{0< i\leqslant j<m-1}
			 A{\scriptsize\left(\begin{array}{rrrrrrrr}
			 	-u_1-\cdots-u_m,& u_1,& \dots,& u_{i-1},& u_i+\cdots+u_{j+1},& u_{j+2},& \dots,& u_{m-1} \\
			 	\sigma_m^{-1},& \sigma_1\sigma_m^{-1},& \dots,& \sigma_{i-1}\sigma_m^{-1},& \sigma_{j+1}\sigma_m^{-1},& \sigma_{j+2}\sigma_m^{-1},& \dots,& \sigma_{m-1}\sigma_m^{-1}
			\end{array}\right)} \\
			&\hspace{5cm}\cdot B{\scriptsize\left(\begin{array}{rrr}
				u_i,& \dots,& u_j \\
				\sigma_i\sigma_{j+1}^{-1},& \dots,& \sigma_j\sigma_{j+1}^{-1} \\
			\end{array}\right)} \\
		&-\sum_{1< i\leqslant j\leqslant m-1}
			 A{\scriptsize\left(\begin{array}{rrrrrrrr}
			 	-u_1-\cdots-u_m,& u_1,& \dots,& u_{i-2},& u_{i-1}+\cdots+u_{j},& u_{j+1},& \dots,& u_{m-1} \\
			 	\sigma_m^{-1},& \sigma_1\sigma_m^{-1},& \dots,& \sigma_{i-2}\sigma_m^{-1},& \sigma_{i-1}\sigma_m^{-1},& \sigma_{j+1}\sigma_m^{-1},& \dots,& \sigma_{m-1}\sigma_m^{-1}
			\end{array}\right)} \\
			&\hspace{5cm}\cdot B{\scriptsize\left(\begin{array}{rrr}
				u_i,& \dots,& u_j \\
				\sigma_i\sigma_{i-1}^{-1},& \dots,& \sigma_j\sigma_{i-1}^{-1} \\
			\end{array}\right)}.
		\intertext{Because $ A, B$ are $\mathrm{push}$-invariant, we get}
		=&\sum_{0\leqslant j<m-1}
			 A{\scriptsize\left(\begin{array}{rrr}
			 	u_{j+2},& \dots,& u_m \\
			 	\sigma_{j+2}\sigma_{j+1}^{-1},& \dots,& \sigma_{m}\sigma_{j+1}^{-1}
			 \end{array}\right)}
			 B{\scriptsize\left(\begin{array}{rrrr}
			 	u_1,& \dots,& u_j,& u_{j+1}+\cdots+u_m \\
			 	\sigma_1,& \dots,& \sigma_j,& \sigma_{j+1}
			 \end{array}\right)} \\
		&-\sum_{1\leqslant j\leqslant m-1}
			 A{\scriptsize\left(\begin{array}{rrr}
			 	u_{j+1},& \dots,& u_m \\
			 	\sigma_{j+1},& \dots,& \sigma_{m}
			 \end{array}\right)}
			 B{\scriptsize\left(\begin{array}{rrr}
				u_1,& \dots,& u_j \\
				\sigma_1,& \dots,& \sigma_j \\
			\end{array}\right)} \\
		&+\sum_{0< i\leqslant j<m-1}
			A{\scriptsize\left(\begin{array}{rrrcrrr}
				u_1,& \dots,& u_{i-1},& u_i+\cdots+u_{j+1},& u_{j+2},& \dots,& u_m \\
				\sigma_1,& \dots,& \sigma_{i-1},& \sigma_{j+1},& \sigma_{j+2},& \dots,& \sigma_m
			\end{array}\right)}
			B{\scriptsize\left(\begin{array}{rrr}
				u_i,& \dots,& u_j \\
				\sigma_i\sigma_{j+1}^{-1},& \dots,& \sigma_j\sigma_{j+1}^{-1}
			\end{array}\right)} \\
		&-\sum_{1< i\leqslant j\leqslant m-1}
			 A{\scriptsize\left(\begin{array}{rrrcrrr}
				u_1,& \dots,& u_{i-2},& u_{i-1}+\cdots+u_j,& u_{j+1},& \dots,& u_m \\
				\sigma_1,& \dots,& \sigma_{i-2},& \sigma_{j-1},& \sigma_{j+1},& \dots,& \sigma_m
			\end{array}\right)}
			 B{\scriptsize\left(\begin{array}{rrr}
				u_i,& \dots,& u_j \\
				\sigma_i\sigma_{i-1}^{-1},& \dots,& \sigma_j\sigma_{i-1}^{-1} \\
			\end{array}\right)} \\
		=&\sum_{\substack{\omega=\alpha\beta \\
			\alpha,\beta\neq\emptyset}}
			 A(\lrflex{\alpha}{\beta})
			 B(\ulflex{\alpha}{\beta})
		-( B\times A)\varia{u_1,\ \dots,\ u_m}{\sigma_1,\ \dots,\ \sigma_m} \\
		&+\sum_{\substack{\omega=\alpha\beta\gamma \\
			\beta,\gamma\neq\emptyset}}
			 A(\alpha\urflex{\beta}{\gamma})
			 B(\llflex{\beta}{\gamma})
		-\sum_{0<i\leqslant m-1} A{\scriptsize\left(\begin{array}{rrrr}
			u_1,& \dots,& u_{i-1},& u_i+\cdots+u_m \\
			\sigma_1,& \dots,& \sigma_{i-1},& \sigma_m
			\end{array}\right)} \\
			&\hspace{8.5cm}\cdot B{\scriptsize\left(\begin{array}{rrr}
				u_i,& \dots,& u_{m-1} \\
				\sigma_i\sigma_{m}^{-1},& \dots,& \sigma_j\sigma_{m}^{-1}
			\end{array}\right)} \\
		&-\left\{\sum_{\substack{\omega=\alpha\beta\gamma \\
			\alpha,\beta\neq\emptyset}}
			 A(\ulflex{\alpha}{\beta}\gamma)
			 B(\lrflex{\alpha}{\beta})\right.
		-\left.\sum_{\substack{\omega=\alpha\beta \\
			\alpha,\beta\neq\emptyset}}
			 A(\ulflex{\alpha}{\beta})
			 B(\lrflex{\alpha}{\beta})\right\}.
	\end{align*}
	So we get
	\begin{align}\label{push on arit}
	 \mathrm{push}&(\mathrm{arit}_u( B)( A))(\omega) \\
		=&(\mathrm{arit}_u( B)( A))(\omega) \nonumber
		-( B\times A)(\omega) \\
		&-\sum_{i=1}^{m-1} A{\scriptsize\left(\begin{array}{rrrr}
				u_1,& \dots,& u_{i-1},& u_i+\cdots+u_m \\
				\sigma_1,& \dots,& \sigma_{i-1},& \sigma_m
			\end{array}\right)}
			B{\scriptsize\left(\begin{array}{rrr}
				u_i,& \dots,& u_{m-1} \\
				\sigma_i\sigma_{m}^{-1},& \dots,& \sigma_j\sigma_{m}^{-1}
			\end{array}\right)} \nonumber\\
		&+\sum_{\substack{\omega=\alpha\beta \nonumber\\
			\alpha,\beta\neq\emptyset}}
			\bigl\{ A(\ulflex{\alpha}{\beta})
			 B(\lrflex{\alpha}{\beta})
		+ A(\lrflex{\alpha}{\beta})
		 B(\ulflex{\alpha}{\beta})\bigr\}. \nonumber
	\end{align}
	Therefore, by using \eqref{push on product} and \eqref{push on arit}, we have
	\begin{align*}
		\mathrm{push}(\mathrm{ari}_u( A, B))(\omega)
		=&\mathrm{push}(\mathrm{arit}_u( B)( A))(\omega)-\mathrm{push}(\mathrm{arit}_u( A)( B))(\omega) \\
		&\quad +\mathrm{push}( A\times  B)(\omega)-\mathrm{push}( B\times  A)(\omega) \\
		=&(\mathrm{arit}_u( B)( A))(\omega)-(\mathrm{arit}_u( A)( B))(\omega) \\
		&\quad +( A\times  B)(\omega)-( B\times  A)(\omega) \\
		=&\mathrm{ari}_u( A, B)(\omega).
	\end{align*}
	Thus $\mathrm{ari}_u( A, B)$ is $\mathrm{push}$-invariant.
\end{proof}

\begin{rem}
This proposition improves the results of \cite[\S 4.1.3]{RS},
which shows that the intersection $\mathrm{ARI}_{\mathrm{push}}(\Gamma)\cap\mathrm{ARI}_\mathrm{al}^\mathrm{pol}(\Gamma)$
forms a Lie algebra under the $\mathrm{ari}_u$-bracket when $\Gamma=\{e\}$.
\end{rem}

Secondly we show that the set of pus-neutral moulds is closed under the $\mathrm{ari}_v$-bracket.
The following proposition is a generalization of \cite[Lemma 21]{RS} which treats the case when $\Gamma =\{e\}$.

\begin{prop}\label{pus-neutral under ari}
	If $ A, B\in\overline{\mathrm{ARI}}(\Gamma)$ are $\mathrm{pus}$-neutral, then $\mathrm{ari}_v( A, B)$ is $\mathrm{pus}$-neutral.
\end{prop}

\begin{proof}
The proof goes in the same way to that of \cite{RS}.
Let $m\geqslant1$.
Since the algebra $\overline{\mathrm{ARI}}(\Gamma)$ is graded by depth,
it is enough to prove  $A$ and $B\in \mathrm{ARI}(\Gamma)$ with
depth $k$ and $l\in\mathbb{N}$ with $m=k+l$.

	Firstly, we prove that $[ A, B]$ is $\mathrm{pus}$-neutral. We calculate
	\begin{align*}
		&\sum_{i=1}^{m}\mathrm{pus}^i([ A, B])^m\varia{\sigma_1,\ \dots,\ \sigma_m}{v_1,\ \dots,\ v_m}
		=\sum_{i\in\mathbb{Z}/m\mathbb{Z}}( A\times  B- B\times  A)^m\varia{\sigma_{1+i},\ \dots,\ \sigma_{m+i}}{v_{1+i},\ \dots,\ v_{m+i}} \\
		&=\sum_{i\in\mathbb{Z}/m\mathbb{Z}}
		\left\{
			A^k\varia{\sigma_{1+i},\ \dots,\ \sigma_{k+i}}{v_{1+i},\ \dots,\ v_{k+i}}
			B^l\varia{\sigma_{k+1+i},\ \dots,\ \sigma_{m+i}}{v_{k+1+i},\ \dots,\ v_{m+i}}
			-B^l\varia{\sigma_{1+i},\ \dots,\ \sigma_{l+i}}{v_{1+i},\ \dots,\ v_{l+i}}
			A^k\varia{\sigma_{l+1+i},\ \dots,\ \sigma_{m+i}}{v_{l+1+i},\ \dots,\ v_{m+i}}
		\right\} \\
		&=\sum_{i\in\mathbb{Z}/m\mathbb{Z}}
		\left\{
			A^k\varia{\sigma_{1+i},\ \dots,\ \sigma_{k+i}}{v_{1+i},\ \dots,\ v_{k+i}}
			B^l\varia{\sigma_{k+1+i},\ \dots,\ \sigma_{m+i}}{v_{k+1+i},\ \dots,\ v_{m+i}}
			-B^l\varia{\sigma_{1+i-l},\ \dots,\ \sigma_{i}}{v_{1+i-l},\ \dots,\ v_{i}}
			A^k\varia{\sigma_{1+i},\ \dots,\ \sigma_{k+i}}{v_{1+i},\ \dots,\ v_{k+i}}
		\right\} \\
		&=\sum_{i\in\mathbb{Z}/m\mathbb{Z}}
		\left\{
			A^k\varia{\sigma_{1+i},\ \dots,\ \sigma_{k+i}}{v_{1+i},\ \dots,\ v_{k+i}}
			B^l\varia{\sigma_{k+1+i},\ \dots,\ \sigma_{m+i}}{v_{k+1+i},\ \dots,\ v_{m+i}}
			-B^l\varia{\sigma_{k+1+i},\ \dots,\ \sigma_{m+i}}{v_{k+1+i},\ \dots,\ v_{m+i}}
			A^k\varia{\sigma_{1+i},\ \dots,\ \sigma_{k+i}}{v_{1+i},\ \dots,\ v_{k+i}}
		\right\}
	\end{align*}
	The first and the second terms are equal, so $[ A, B]$ is $\mathrm{pus}$-neutral.

	Secondly, we prove that $\mathrm{arit}_v( B)( A)$ is $\mathrm{pus}$-neutral. Then we have
	\begin{align*}
		&(\mathrm{arit}_v( B)( A))^m\varia{\sigma_1,\ \dots,\ \sigma_m}{v_1,\ \dots,\ v_m}
		=\sum_{\substack{\omega=\alpha\beta\gamma \\
		\beta,\gamma\neq\emptyset}}
		 A^{k}(\alpha\urflex{\beta}{\gamma}) B^{l}(\llflex{\beta}{\gamma})
		-\sum_{\substack{\omega=\alpha\beta\gamma \\
		\alpha,\beta\neq\emptyset}}
		 A^{k}(\ulflex{\alpha}{\beta}\gamma) B^{l}(\lrflex{\alpha}{\beta}).
	\end{align*}
	Because $l\geqslant1$ and $B$ is with depth $l$, we may put
	\footnote{For the first (resp. second) term, the word $\alpha$ (resp. $\gamma$) can be $\emptyset$ and the word $\gamma$ (resp. $\alpha$) is not $\emptyset$. So the index $j$ must run from $0$ (resp. $1$) to $k-1=m-l-1$ (resp. $k=m-l$).}
	$\alpha=\varia{\sigma_1,\ \dots,\ \sigma_{j}}{v_1,\ \dots,\ v_{j}}$,
	$\beta=\varia{\sigma_{j+1},\ \dots,\ \sigma_{j+l}}{v_{j+1},\ \dots,\ v_{j+l}}$ and
	$\gamma=\varia{\sigma_{j+l+1},\ \dots,\ \sigma_{m}}{v_{j+l+1},\ \dots,\ v_{m}}$ and we get
	\begin{align*}
		&(\mathrm{arit}_v( B)( A))^m\varia{\sigma_1,\ \dots,\ \sigma_m}{v_1,\ \dots,\ v_m} \\
		&=\sum_{j=0}^{k-1}
		A^{k}{\scriptsize\left(\begin{array}{rrrcrrr}
			\sigma_1,& \dots,& \sigma_{j},& \sigma_{j+1}\cdots\sigma_{j+l+1},& \sigma_{j+l+2},& \dots,& \sigma_{m} \\
			v_1,& \dots,& v_{j},& v_{j+l+1},& v_{j+l+2},& \dots,& v_{m}
		\end{array}\right)}
		B^{l}{\scriptsize\left(\begin{array}{rrr}
			\sigma_{j+1},& \dots,& \sigma_{j+l} \\
			v_{j+1}-v_{j+l+1},& \dots,& v_{j+l}-v_{j+l+1} \\
		\end{array}\right)} \\
		&\quad-\sum_{j=1}^{k}
		A^{k}{\scriptsize\left(\begin{array}{rrrcrrr}
		 	\sigma_1,& \dots,& \sigma_{j-1},& \sigma_{j}\cdots\sigma_{j+l},& \sigma_{j+l+1},& \dots,& \sigma_{m} \\
		 	v_1,& \dots,& v_{j-1},& v_{j},& v_{j+l+1},& \dots,& v_{m}
		\end{array}\right)}
		B^{l}{\scriptsize\left(\begin{array}{rrr}
			\sigma_{j+1},& \dots,& \sigma_{j+l} \\
			v_{j+1}-v_{j},& \dots,& v_{j+l}-v_{j} \\
		\end{array}\right)}.
	\end{align*}
	Here, for $0\leqslant j\leqslant k-1$, we have
	\begin{align*}
		&\sum_{i\in\mathbb{Z}/m\mathbb{Z}}\sum_{j=0}^{k-1}
		A^{k}{\scriptsize\left(\begin{array}{rrrcrrr}
			\sigma_{i+1},& \dots,& \sigma_{i+j},& \sigma_{i+j+1}\cdots\sigma_{i+j+l+1},& \sigma_{i+j+l+2},& \dots,& \sigma_{i+m} \\
			v_{i+1},& \dots,& v_{i+j},& v_{i+j+l+1},& v_{i+j+l+2},& \dots,& v_{i+m}
		\end{array}\right)} \\
		&\hspace{5cm}\cdot
		B^{l}{\scriptsize\left(\begin{array}{rrr}
			\sigma_{i+j+1},& \dots,& \sigma_{i+j+l} \\
			v_{i+j+1}-v_{i+j+l+1},& \dots,& v_{i+j+l}-v_{i+j+l+1} \\
		\end{array}\right)} \\
		&=\sum_{i\in\mathbb{Z}/m\mathbb{Z}}\sum_{j=0}^{k-1}
		A^{k}{\scriptsize\left(\begin{array}{rrrcrrr}
			\sigma_{i+1-j},& \dots,& \sigma_{i},& \sigma_{i+1}\cdots\sigma_{i+l+1},& \sigma_{i+l+2},& \dots,& \sigma_{i+m-j} \\
			v_{i+1-j},& \dots,& v_{i},& v_{i+l+1},& v_{i+l+2},& \dots,& v_{i+m-j}
		\end{array}\right)} \\
		&\hspace{5cm}\cdot
		B^{l}{\scriptsize\left(\begin{array}{rrr}
			\sigma_{i+1},& \dots,& \sigma_{i+l} \\
			v_{i+1}-v_{i+l+1},& \dots,& v_{i+l}-v_{i+l+1} \\
		\end{array}\right)}. \\
		&=\sum_{i\in\mathbb{Z}/m\mathbb{Z}}\sum_{j=0}^{k-1}
		(\mathrm{pus}^j( A))^{k}{\scriptsize\left(\begin{array}{crrr}
			\sigma_{i+1}\cdots\sigma_{i+l+1},& \sigma_{i+l+2},& \dots,& \sigma_{i+m} \\
			v_{i+l+1},& v_{i+l+2},& \dots,& v_{i+m}
		\end{array}\right)} \\
		&\hspace{5cm}\cdot
		B^l{\scriptsize\left(\begin{array}{rrr}
			\sigma_{i+1},& \dots,& \sigma_{i+l} \\
			v_{i+1}-v_{i+l+1},& \dots,& v_{i+l}-v_{i+l+1} \\
		\end{array}\right)}.
	\end{align*}
	Since $ A$ is $\mathrm{pus}$-neutral, this is 0.
Similarly by using the $\mathrm{pus}$-neutrality of $ A$, we obtain
	\begin{align*}
		&\sum_{i\in\mathbb{Z}/m\mathbb{Z}}\sum_{j=1}^{k}
		A^{k}{\scriptsize\left(\begin{array}{rrrcrrr}
		 	\sigma_{i+1},& \dots,& \sigma_{i+j-1},& \sigma_{i+j}\cdots\sigma_{i+j+l},& \sigma_{i+j+l+1},& \dots,& \sigma_{i+m} \\
		 	v_{i+1},& \dots,& v_{i+j-1},& v_{i+j},& v_{i+j+l+1},& \dots,& v_{i+m}
		\end{array}\right)} \\
		&\hspace{5cm}\cdot
		B^{l}{\scriptsize\left(\begin{array}{rrr}
			\sigma_{i+j+1},& \dots,& \sigma_{i+j+l} \\
			v_{i+j+1}-v_{i+j},& \dots,& v_{i+j+l}-v_{i+j} \\
		\end{array}\right)}=0
	\end{align*}
	for all $1\leqslant j\leqslant k$.
	Therefore, we get
	\begin{align*}
       \sum_{i=1}^{m}\mathrm{pus}^i(\mathrm{arit}_v( B)( A))^m\varia{\sigma_1,\ \dots,\ \sigma_m}{v_1,\ \dots,\ v_m}
       =\sum_{i\in\mathbb{Z}/m\mathbb{Z}}(\mathrm{arit}_v( B)( A))^m\varia{\sigma_{1+i},\ \dots,\ \sigma_{m+i}}{v_{1+i},\ \dots,\ v_{m+i}}=0.
	\end{align*}
Whence  $\mathrm{arit}_v( B)( A)$ is $\mathrm{pus}$-neutral.

	Thirdly,  in the same way, we can show that $\mathrm{arit}_v( A)( B)$ is $\mathrm{pus}$-neutral by using the $\mathrm{pus}$-neutrality of $ B$.

	Thus we complete the proof because  $\mathrm{ari}_v( A, B)=[ A, B]+\mathrm{arit}_v( B)( A)-\mathrm{arit}_v( A)( B)$.
\end{proof}

We need the following lemma for the proof of Theorem \ref{thm: ARIpushpusnu Lie}.

\begin{lem
\label{swap-ari commutation}
	If $ A, B\in\mathrm{ARI}(\Gamma)$ are $\mathrm{push}$-invariant, then we have
	$$\mathrm{swap}(\mathrm{ari}_u( A, B))=\mathrm{ari}_v(\mathrm{swap}( A),\mathrm{swap}( B)).$$
\end{lem}

\begin{proof}
This  follows from the proof of  \cite[Lemma 2.4.1]{S-ARIGARI},
which actually works for $\mathrm{BARI}$.
\end{proof}

\begin{thm}\label{thm: ARIpushpusnu Lie}
	The set $\mathrm{ARI}(\Gamma)_{\mathrm{push}/ \mathrm{pusnu}}$ forms a Lie subalgebra of $\mathrm{ARI}(\Gamma)_{\mathrm{push}}$ under the $\mathrm{ari}_u$-bracket.
\end{thm}

\begin{proof}
    Let $ A, B \in \mathrm{ARI}(\Gamma)_{\mathrm{push}/\mathrm{pusnu}}$. Then by Proposition \ref{prop:ARIpush Lie},
    $\mathrm{ari}_u( A, B)\in \mathrm{ARI}(\Gamma)_\mathrm{push}$.
	Let $m\geqslant1$. Because $ A$ and $ B$ are $\mathrm{push}$-invariant, by Lemma \ref{swap-ari commutation}, we have
	\begin{align*}
		\sum_{i\in\mathbb{Z}/m\mathbb{Z}}\mathrm{pus}^i&\circ\mathrm{swap}(\mathrm{ari}_u( A, B))^m
			\varia{\sigma_1,\ \dots,\ \sigma_m}{v_1,\ \dots,\ v_m} \\
		=&\sum_{i\in\mathbb{Z}/m\mathbb{Z}}\mathrm{pus}^i\bigl(\mathrm{ari}_v(\mathrm{swap}( A),\mathrm{swap}( B))\bigr)^m
			\varia{\sigma_1,\ \dots,\ \sigma_m}{v_1,\ \dots,\ v_m}.
	\end{align*}
	By Proposition \ref{pus-neutral under ari}, the right hand side is equal to 0. Hence $\mathrm{swap}(\mathrm{ari}_u( A, B))$ is pus-neutral.
	Thus we obtain $\mathrm{ari}_u(A, B)\in \mathrm{ARI}(\Gamma)_{\mathrm{push}/\mathrm{pusnu}}$.
\end{proof}

\begin{rem}
The above theorem improves \cite[Corollary 22]{RS} which shows that
$\mathrm{ARI}(\Gamma)_{\mathrm{push}/ \mathrm{pusnu}}\cap \mathrm{ARI}_\mathrm{al}^\mathrm{pol}(\Gamma)$
forms a Lie algebra  when $\Gamma=\{e\}$.
\end{rem}

Our Lie  algebra $\mathrm{ARI}(\Gamma)_{\mathrm{push}/ \mathrm{pusnu}}$
will play an important role in the following sections.

\begin{defn}\label{def:ARID}
For $N\geqslant 1$, put $\Gamma^N:=\{g^N\in\Gamma\ |\  g\in\Gamma\}$.
We consider the map $i_N:\mathrm{ARI}(\Gamma)\to\mathrm{ARI}(\Gamma^N)$
which  is  given by
\begin{equation}
i_N(M)^m\varia{x_1,\ \dots,\ x_m}{\sigma_1,\ \dots,\ \sigma_m}
=
M^m\varia{x_1,\ \dots,\ x_m}{\sigma_1,\ \dots,\ \sigma_m}
\end{equation}
and also the map $m_N:\mathrm{ARI}(\Gamma)\to\mathrm{ARI}(\Gamma^N)$
which  is  given by
\begin{equation}
m_N(M)^m\varia{x_1,\ \dots,\ x_m}{\sigma_1,\ \dots,\ \sigma_m}

\sum_{\tau_i^N=\sigma_i}
M^m\varia{Nx_1,\ \dots,\ Nx_m}{\quad \tau_1,\ \dots,\ \tau_m}.
\end{equation}
We define  the following  $\mathbb Q$-linear subspaces
which are subject to the distribution relations:
\begin{align*}
&\mathrm{ARID}(\Gamma):=\{M\in\mathrm{ARI}(\Gamma)\ |\ i_N(M)=m_N(M)
\text{ for all } N\geqslant 1
\text{ with }
N\bigm |  |\Gamma|
\}, \\
&\mathrm{ARID}(\Gamma)_{\underline\mathrm{al}/\underline\mathrm{al}}:=\mathrm{ARID}(\Gamma)\cap
\mathrm{ARI}(\Gamma)_{\underline\mathrm{al}/\underline\mathrm{al}}.
\end{align*}
\end{defn}

\begin{prop} \label{prop:ARID Lie algebra}
The $\mathbb Q$-linear subspace $\mathrm{ARID}(\Gamma)$ forms a filtered Lie subalgebra of
$\mathrm{ARI}(\Gamma)$ under the $\mathrm{ari}_u$-bracket.
\end{prop}

\begin{proof}
First we prove that $m_N$ is a Lie algebra homomorphism, that is,
\begin{equation*}
	m_N(\mathrm{ari}_u(A,B))=\mathrm{ari}_u(m_N(A),m_N(B))
\end{equation*}
for $A,B\in\mathrm{ARI}(\Gamma)$. Let $m\geqslant0$. We have
\begin{align*}
	m_N(A\times B)^m({\bf x}_m)
	&= \sum_{\substack{
		\tau_i^N=\sigma_i \\
		1\leqslant i\leqslant m}}
	(A\times B)^m\varia{Nx_1,\ \dots,\ Nx_m}{\quad \tau_1,\ \dots,\ \tau_m} \\
	&= \sum_{\substack{
		\tau_i^N=\sigma_i \\
		1\leqslant i\leqslant m}}
	\sum_{j=0}^m
	A^i\varia{Nx_1,\ \dots,\ Nx_j}{\quad \tau_1,\ \dots,\ \tau_j}
	B^{m-i}\varia{Nx_{j+1},\ \dots,\ Nx_m}{\quad \tau_{j+1},\ \dots,\ \tau_m} \\
	&= \sum_{j=0}^m
	\left\{
	\sum_{\substack{
		\tau_i^N=\sigma_i \\
		1\leqslant i\leqslant j}}A^i\varia{Nx_1,\ \dots,\ Nx_j}{\quad \tau_1,\ \dots,\ \tau_j}
	\right\}
	\left\{
	\sum_{\substack{
		\tau_i^N=\sigma_i \\
		j+1\leqslant i\leqslant m}}	B^{m-i}\varia{Nx_{j+1},\ \dots,\ Nx_m}{\quad \tau_{j+1},\ \dots,\ \tau_m}
	\right\} \\
	&= \sum_{j=0}^m
	m_N(A)^i\varia{x_1,\ \dots,\ x_j}{\sigma_1,\ \dots,\ \sigma_j}
	m_N(B)^{m-i}\varia{x_{j+1},\ \dots,\ x_m}{\sigma_{j+1},\ \dots,\ \sigma_m} \\
	&= m_N(A\times B)^m({\bf x}_m).
\end{align*}
Similarly, we have
\begin{align*}
	&m_N(\mathrm{arit}_u(B)(A))^m({\bf x}_m) \\
	&=\sum_{\substack{
		\nu_i^N=\sigma_i \\
		1\leqslant i\leqslant m}}
	\mathrm{arit}(B)(A)^m\varia{Nx_1,\ \dots,\ Nx_m}{\quad \nu_1,\ \dots,\ \nu_m} \\
	&= \sum_{\substack{
		\nu_i^N=\sigma_i \\
		1\leqslant i\leqslant m}}
	\left\{
	\sum_{1\leqslant k\leqslant l<m}
	A{\scriptsize\left(\begin{array}{rrrcrrr}
	 	Nx_1,& \dots,& Nx_{k-1},& N(x_k+\cdots+x_{l+1}),& Nx_{l+2},& \dots,& Nx_{m} \\
	 	\nu_1,& \dots,& \nu_{k-1},& \nu_{l+1},& \nu_{l+2},& \dots,& \nu_{m}
	\end{array}\right)}
	\right. \\
	&\hspace{5cm}\cdot \left.
	\vphantom{\sum_{1\leqslant k\leqslant l<m}}
	B{\scriptsize\left(\begin{array}{rrr}
		Nx_k,& \dots,& Nx_l \\
		\nu_k\nu_{l+1}^{-1},& \dots,&\nu_l\nu_{l+1}^{-1} \\
	\end{array}\right)}
	\right\} \\
	&\quad -\sum_{\substack{
		\nu_i^N=\sigma_i \\
		1\leqslant i\leqslant m}}
		\left\{
	\sum_{1< k\leqslant l\leqslant m}
	A{\scriptsize\left(\begin{array}{rrrcrrr}
	 	Nx_1,& \dots,& Nx_{k-2},& N(x_{k-1}+\cdots+x_{l}),& Nx_{l+1},& \dots,& Nx_{m} \\
	 	\nu_1,& \dots,& \nu_{k-2},& \tau_{k-1},& \nu_{l+1},& \dots,& \nu_{m}
	\end{array}\right)}
	\right. \\
	&\hspace{5cm}\cdot \left.
	\vphantom{\sum_{1\leqslant k\leqslant l<m}}
	B{\scriptsize\left(\begin{array}{rrr}
		Nx_k,& \dots,& Nx_l \\
		\nu_k\nu_{k-1}^{-1},& \dots,& \nu_l\nu_{k-1}^{-1} \\
	\end{array}\right)}
	\right\}.
\end{align*}
Here, we put $S_{k,l}:=\{i\in\mathbb{N}\ |\ 1\leqslant i\leqslant k-1\}\cup\{i\in\mathbb{N}\ |\ l+1\leqslant i\leqslant m\}$ for $1\leqslant k\leqslant l\leqslant m$. Then we can divide the set of variables of the summation in the following :
\begin{equation*}
	\{\nu_i\ |\ 1\leqslant i\leqslant m\}
	=\{\nu_i\ |\ i\in S_{k,l}\}
	\cup\{\nu_i\ |\ k\leqslant i\leqslant l\}.
\end{equation*}
So by substituting $\nu_i:=\tau_i$ for $i\in S_{k,l}$ and $\nu_i:=\tau_i\tau_{l+1}$ for $k\leqslant i\leqslant l$ for the first summation, and by substituting $\nu_i:=\tau_i$ for $i\in S_{k,l}$ and $\nu_i:=\tau_i\tau_{k-1}$ for $k\leqslant i\leqslant l$ for the second summation, we calculate
\begin{align*}
	&m_N(\mathrm{arit}_u(B)(A))^m({\bf x}_m) \\
	&=  \sum_{1\leqslant k\leqslant l<m}
	\left\{
	\sum_{\substack{
		\tau_i^N=\sigma_i \\
		i\in S_{k,l}}}
	A{\scriptsize\left(\begin{array}{rrrcrrr}
	 	Nx_1,& \dots,& Nx_{k-1},& N(x_k+\cdots+x_{l+1}),& Nx_{l+2},& \dots,& Nx_{m} \\
	 	\tau_1,& \dots,& \tau_{k-1},& \tau_{l+1},& \tau_{l+2},& \dots,& \tau_{m}
	\end{array}\right)}
	\right\} \\
	&\hspace{5cm}\cdot \left\{
	\sum_{\substack{
		\tau_i^N=\sigma_i\sigma_{l+1}^{-1} \\
		k\leqslant i\leqslant l}}
	B{\scriptsize\left(\begin{array}{crc}
		Nx_k,& \dots,& Nx_l \\
		\tau_k,& \dots,&\tau_l \\
	\end{array}\right)}
	\right\} \\
	&\quad -\sum_{1< k\leqslant l\leqslant m}
	\left\{
	\sum_{\substack{
		\tau_i^N=\sigma_i \\
		i\in S_{k,l}}}
	A{\scriptsize\left(\begin{array}{rrrcrrr}
	 	Nx_1,& \dots,& Nx_{k-2},& N(x_{k-1}+\cdots+x_{l}),& Nx_{l+1},& \dots,& Nx_{m} \\
	 	\tau_1,& \dots,& \tau_{k-2},& \tau_{k-1},& \tau_{l+1},& \dots,& \tau_{m}
	\end{array}\right)}
	\right\} \\
	&\hspace{5cm}\cdot \left\{
	\sum_{\substack{
		\tau_i^N=\sigma_i\sigma_{k-1}^{-1} \\
		k\leqslant i\leqslant l}}
	B{\scriptsize\left(\begin{array}{crc}
		Nx_k,& \dots,& Nx_l \\
		\tau_k,& \dots,& \tau_l \\
	\end{array}\right)}
	\right\} \\
	&= \sum_{1\leqslant k\leqslant l<m}
	m_N(A){\scriptsize\left(\begin{array}{rrrcrrr}
	 	x_1,& \dots,& x_{k-1},& x_k+\cdots+x_{l+1},& x_{l+2},& \dots,& x_{m} \\
	 	\sigma_1,& \dots,& \sigma_{k-1},& \sigma_{l+1},& \sigma_{l+2},& \dots,& \sigma_{m}
	\end{array}\right)} \\
	&\hspace{5cm}\cdot m_N(B){\scriptsize\left(\begin{array}{rrr}
		x_k,& \dots,& x_l \\
		\sigma_k\sigma_{l+1}^{-1},& \dots,&\sigma_l\sigma_{l+1}^{-1} \\
	\end{array}\right)} \\
	&\quad -\sum_{1< k\leqslant l\leqslant m}
	m_N(A){\scriptsize\left(\begin{array}{rrrcrrr}
	 	x_1,& \dots,& x_{k-2},& x_{k-1}+\cdots+x_{l},& x_{l+1},& \dots,& x_{m} \\
	 	\sigma_1,& \dots,& \sigma_{k-2},& \sigma_{k-1},& \sigma_{l+1},& \dots,& \sigma_{m}
	\end{array}\right)} \\
	&\hspace{5cm}\cdot m_N(B){\scriptsize\left(\begin{array}{rrr}
		x_k,& \dots,& x_l \\
		\sigma_k\sigma_{k-1}^{-1},& \dots,& \sigma_l\sigma_{k-1}^{-1} \\
	\end{array}\right)}. \\
	\intertext{By putting $\alpha=\varia{x_1,\ \dots,\ x_{k-1}}{\sigma_1,\ \dots,\ \sigma_{k-1}}$, $\beta=\varia{x_k,\ \dots,\ x_l}{\sigma_k,\ \dots,\ \sigma_l}$ and $\gamma=\varia{x_{l+1},\ \dots,\ x_m}{\sigma_{l+1},\ \dots,\ \sigma_m}$, we get}
	&= \sum_{\substack{{\bf x}_m=\alpha\beta\gamma \\
		\beta,\gamma\neq\emptyset}}
	m_N(A)(\alpha\urflex{\beta}{\gamma})
	m_N(B)(\llflex{\beta}{\gamma})
	-\sum_{\substack{{\bf x}_m=\alpha\beta\gamma \\
		\alpha,\beta\neq\emptyset}}
	m_N(A)(\ulflex{\alpha}{\beta}\gamma)
	m_N(B)(\lrflex{\alpha}{\beta}) \\
	&= \mathrm{arit}_u(m_N(B))(m_N(A))^m({\bf x}_m).
\end{align*}
Hence we get
\begin{align*}
	m_N(\mathrm{ari}_u(A,B))
	&= m_N(\mathrm{arit}_u(B)(A))-m_N(\mathrm{arit}_u(A)(B))+m_N(A\times B)-m_N(B\times A) \\
	&= \mathrm{arit}_u(m_N(B))(m_N(A))-\mathrm{arit}_u(m_N(A))(m_N(B)) \\
	&\hspace{1cm}+m_N(A)\times m_N(B)-m_N(B)\times m_N(A) \\
	&= \mathrm{ari}_u(m_N(A),m_N(B)).
\end{align*}
Therefore, $m_N$ is a Lie algebra homomorphism.

It is obvious that the map $i_N$ is a Lie algebra homomorphism.
Since both $m_N$ and $i_N$ are Lie algebra homomorphisms,
$\ker(i_N-m_N)$ forms Lie algebra.

By the definition of $\mathrm{ARID}(\Gamma)$, we have
$\mathrm{ARID}(\Gamma)=\underset{N\bigm |  |\Gamma|}{\bigcap}\ker(i_N-m_N)$.
Therefore $\mathrm{ARID}(\Gamma)$ forms a Lie subalgebra under the $\mathrm{ari}_u$-bracket.
\end{proof}

\begin{cor}\label{cor:ARID alal Lie algebra}
The $\mathbb Q$-linear subspace $\mathrm{ARID}(\Gamma)_{\underline\mathrm{al}/\underline\mathrm{al}}$ forms a filtered Lie subalgebra of
$\mathrm{ARI}(\Gamma)$ under the $\mathrm{ari}_u$-bracket.
\end{cor}

\begin{proof}
It is a direct consequence of Proposition \ref{ARIalal Lie algebra} and
Proposition \ref{prop:ARID Lie algebra}.
\end{proof}

\section{Kashiwara-Vergne Lie algebra}\label{sec:Kashiwara-Vergne Lie algebra}

We introduce
the Kashiwara-Vergne bigraded Lie algebra $\mathfrak{lkrv}(\Gamma)_{\bullet\bullet}$
associated with a finite abelian group $\Gamma$
and give its  mould theoretical interpretation by using
$\mathrm{ARI}(\Gamma)_{\mathrm{push}/ \mathrm{pusnu}}$.

\subsection{$\Gamma$-variant of  the KV condition}\label{subsec:KV condition}
We investigate a variant of the defining conditions of  Kashiwara-Vergne graded Lie algebra associated with a finite abelian group $\Gamma$
(cf. Definition \ref{defn: KV condition})
and explain its mould theoretical interpretation in Theorem \ref{thm:reform:krv}.

Let $\mathbb L=\oplus_{w\geqslant 1}\mathbb L_w$ be the free graded Lie $\mathbb Q$-algebra generated by $N+1$ variables $x$ and $y_\sigma$ ($\sigma\in\Gamma$) with $\deg x=\deg y_\sigma=1$.
Here $\mathbb L_w$ is the $\mathbb Q$-linear space generated by Lie monomials whose total degree is $w$.
Occasionally we regard $\mathbb L$ as a bigraded Lie algebra
$\mathbb L_{\bullet\bullet}=\oplus_{w,d}\mathbb L_{w,d}$,
where $\mathbb L_{w,d}$ is the $\mathbb Q$-linear space generated by Lie monomials whose {\it weight} (the total degree) is $w$ and {\it depth} (the degree with respect to all $y_\sigma$) is $d$.
We encode $\mathbb L$ with  a structure of filtered graded  Lie algebra
by the filtration $\mathrm{Fil}_{\mathcal{D}}^d \mathbb L_w:=\oplus_{N\geqslant d}\mathbb L_{w,N}$ for $d>0$
and denote the associated bigraded Lie algebra by
$\mathrm{gr}_{\mathcal{D}}\mathbb L=\oplus_{w,d}\mathrm{gr}^d_{\mathcal{D}}\mathbb L_w$
with $\mathrm{gr}_{\mathcal{D}}^d\mathbb L_w=
\mathrm{Fil}_{\mathcal{D}}^d \mathbb L_w/\mathrm{Fil}_{\mathcal{D}}^{d+1} \mathbb L_w$.

The $N+1$-variable  non-commutative polynomial algebra $\mathbb A=\mathbb Q\langle x,y_\sigma;\sigma\in\Gamma\rangle$ is regarded as
the universal enveloping algebra of $\mathbb L$ and is encoded with the induced degree.
Similarly $\mathbb A$ is encoded with a structure of bigraded algebra;
$\mathbb A_{\bullet\bullet}=\oplus_{w,d}\mathbb A_{w,d}$.
By putting $\mathrm{Fil}_{\mathcal{D}}^d \mathbb A_w:=\oplus_{N\geqslant d}\mathbb A_{w,N}$
for $d>0$,
we also encode $\mathbb A$ with  a structure of filtered  graded algebra.
We define the action of $\tau\in \Gamma$ on $\mathbb A$ (hence of $\mathbb L$) by
$$
\tau(x)=x \text{ and }\tau(y_\sigma)=y_{\tau\sigma}.
$$
For any $h\in\mathbb A$, we denote
$$h=h_xx+\sum_\alpha h_{y_\alpha} y_\alpha=xh^x+\sum_\alpha y_\alpha h^{y_\alpha}.$$

We denote $\pi_Y$ to be  the composition of the natural projection and inclusion:
$$\pi_Y:\mathbb A\twoheadrightarrow \mathbb A/\mathbb A\cdot x
\simeq \mathbb Q\oplus (\oplus_{\sigma\in\Gamma}\mathbb A y_\sigma)
\hookrightarrow\mathbb A$$
and the $\mathbb Q$-linear isomorphism $\mathbf{q}$ on $\mathbb A$ defined by
$$
\mathbf{q}(x^{e_0}y_{\sigma_1}x^{e_1}y_{\sigma_2}\cdots x^{e_{r-1}}y_{\sigma_{r}}x^{e_r})
=x^{e_0}y_{\sigma_1}x^{e_1}y_{\sigma_2\sigma_1^{-1}}\cdots
x^{e_{r-1}} y_{\sigma_r\sigma_{r-1}^{-1}}x^{e_r}
$$
(cf. \cite{R}).
The $\mathbb Q$-linear endomorphism
$$\mathrm{anti}:\mathbb A\to \mathbb A$$
is the palindrome (backwards-writing) operator in $\mathbb A_w$  (cf. \cite[Definition 1.3]{S}).

We put $\mathrm{Cyc}(\mathbb A)$ to be
$\mathbb Q$-linear space generated by cyclic words of $\mathbb A$
and  $\mathrm{tr}:\mathbb A\twoheadrightarrow \mathrm{Cyc}(\mathbb A)$
to be the trace map, the natural projection to
$\mathrm{Cyc}(\mathbb A)$ (cf. \cite{AT}).

\begin{defn
\label{defn: KV condition}
We define
the graded $\mathbb Q$-linear space $\mathfrak{krv}(\Gamma)_\bullet=\oplus_{w> 1}\mathfrak{krv}(\Gamma)_w$,
where
its degree $w$-part $\mathfrak{krv}(\Gamma)_w$ is defined to be the set of Lie elements  $F\in\mathbb L_w$ such that
there exists $G=G(F)$ in $\mathbb L_w$ with
\begin{align}\tag{KV1}
\label{KV1}
&[x,G]+\sum_{\tau\in\Gamma}[y_\tau,\tau(F)]=0,\\
\tag{KV2}
\label{KV2}
& \mathrm{tr}\circ\mathbf{q}\circ\pi_Y(F(z;(y_\sigma)))
=0
 \end{align}
with $z=-x-\sum_{\sigma\in\Gamma} y_\sigma$.
\end{defn}

We note that such $G=G(F)$  uniquely exists when $w>1$.
For $d\geqslant 1$, we put $\mathrm{Fil}_{\mathcal{D}}^d\mathfrak{krv}(\Gamma)_w$ to be the subspace of $\mathfrak{krv}(\Gamma)_w$
consisting of $F\in \mathrm{Fil}_{\mathcal{D}}^d\mathbb L_w$.
By  \eqref{KV2},
$$\mathfrak{krv}(\Gamma)_w=\mathrm{Fil}_{\mathcal{D}}^2\mathfrak{krv}(\Gamma)_w.$$

\begin{lem}
Assume  that $\Gamma=\{e\}$.
Let $F\in\mathbb L_w$ satisfying \eqref{KV1} with $w\geqslant 2$.
Then \eqref{KV2} for $F$ is equivalent to
\begin{equation}\label{eq:original KV2}
\mathrm{tr}(G_xx+F_yy)=0.
\end{equation}
\end{lem}

\begin{proof}
By
$
\mathrm{tr}\circ\mathbf{q}\circ\pi_Y(F(z,y))=\mathrm{tr}\circ\pi_Y(F(z,y))=
\mathrm{tr}(-F_x(z,y)y+F_y(z,y)y),
$
the condition
\eqref{KV2} is equivalent to
$$\mathrm{tr}((F_y-F_x)y)=0.$$

By \eqref{KV1}, we have
$Gx-xG=yF-Fy$.
So $F_y=F^y$ and $G_y=F^x$.
By $F\in\mathbb L_w$, $F_x=(-1)^{w-1}\mathrm{anti}(F^x)$ and
$F_y=(-1)^{w-1}\mathrm{anti}(F^y)$.
By $G\in\mathbb L_w$, we have $\mathrm{tr}(G_xx+G_yy)=0$.
Therefore
$$\mathrm{tr}((F_y-F_x)y)=(-1)^{w-1}\mathrm{tr}((F^y-F^x)y)
=(-1)^{w-1}\mathrm{tr}(F_yy-G_yy)=(-1)^{w-1}\mathrm{tr}(F_yy+G_xx),
$$
whence we get the claim
\end{proof}

\begin{rem}
Since \eqref{eq:original KV2} agrees with original defining condition (KV2)
in \cite{AT},
we see that our $\mathfrak{krv}(\Gamma)_\bullet$ with $\Gamma=\{e\}$
recovers the original Kashiwara-Vergne Lie algebra denoted by $\mathfrak{krv}^0_{2}$
in \cite{AT} and
the depth>1-part of $\mathfrak{krv}$ in \cite[Definition 3]{RS}.
We do not know how their Lie algebras
$\mathfrak{krv}^0_{n+1}$ ($n\geqslant 0$) in \cite{AT} and also
$\mathrm{krv}_{n+1}$ in \cite{AKKN}
are related to our $\mathfrak{krv}(\Gamma)_\bullet$.
It is also not clear if our $\mathfrak{krv}(\Gamma)_\bullet$ forms a Lie algebra or not.
\end{rem}

Let $h\in\mathbb A_w$ be a degree $w$ homogeneous polynomial
with
$h=\sum_{r=0}^wh^r$
and
\begin{equation}\label{eq: word expansion}
h^r=  \sum_{(\sigma_1,\dots,\sigma_r)\in\Gamma^{\oplus r}}\sum_{(e_0,\dots,e_r)\in\mathcal{E}_w^r}
a(h:{}_{\sigma_1,\dots,\sigma_r}^{e_0,\dots,e_r})x^{e_0}y_{\sigma_1}\cdots y_{\sigma_r}x^{e_r}
\in\mathbb A_{w,r}
\end{equation}
where $\mathcal{E}_w^r=\{(e_0,\dots,e_r)\in\mathbb{N}_0^{r+1} \mid \sum_{i=0}^r e_i=w-r \}$.
\begin{defn}\label{def:ma}
By following \cite[Appendix A]{S},
we associate a mould
$$\mathrm{ma}_h=(\mathrm{ma}^0_h,\mathrm{ma}^1_h,\mathrm{ma}^2_h,\dots,\mathrm{ma}^w_h,0,0,\dots) \in\mathcal M(\mathcal F;\Gamma)$$
which is defined by
$\mathrm{ma}^r_h=\{\mathrm{ma}^r_h({}_{\sigma_1,\dots,\sigma_r}^{u_1,\dots,u_r})\}_{(\sigma_1,\dots,\sigma_r)\in\Gamma^{\oplus r}}$
with
\begin{align*}
&
\mathrm{ma}^r_h({}_{\sigma_1,\dots,\sigma_r}^{u_1,\dots,u_r})
=\mathrm{vimo}^r_h({}_{\qquad \sigma_1,\dots,\sigma_r}^{0,u_1,u_1+u_2,\dots,u_1+\cdots+u_r}), \\
&
\mathrm{vimo}^r_h({}^{z_0,\dots,z_r}_{\sigma_1,\dots,\sigma_r})
=\sum_{(e_0,\dots,e_r)\in\mathcal{E}_w^r}
a(h:{}_{\sigma_1^{-1},\dots,\sigma_r^{-1}}^{e_0,\dots,e_r})
z_0^{e_0}z_1^{e_1}z_2^{e_2}\cdots z_r^{e_r}.
\end{align*}
\end{defn}
We start with the following  technical lemma which  is required to our later arguments.

\begin{lem}
When $h\in\mathbb L_{w,r}$, we have
\begin{equation}\label{eq:translation invariance of vimo}
\mathrm{vimo}^r_h({}^{z_0,\dots,z_r}_{\sigma_1,\dots,\sigma_r})
=\mathrm{vimo}^r_h({}^{0,z_1-z_0,\dots,z_r-z_0}_{\qquad \sigma_1,\dots,\sigma_r}),
\end{equation}
\begin{equation}\label{eq:mantar invariance for ma}
\mathrm{mantar}\circ\mathrm{ma}_h^r=\mathrm{ma}_h^r.
\end{equation}
\end{lem}

\begin{proof}
The proof of
\eqref{eq:translation invariance of vimo}
can be done by induction on degree
in the same way to the arguments in
\cite{S} p.71.
Assume $h\in\mathbb L_{w,r}$ is given by $[f,g]$ for some
$f$ and $g\in\mathbb L$ with  depth $s$ and $t$ respectively.
Then by definition, we have
$$
\mathrm{vimo}_h^r({}^{z_0,\dots,z_r}_{\sigma_1,\dots,\sigma_r})
=\mathrm{vimo}_f^s({}^{z_0,\dots,z_s}_{\sigma_1,\dots,\sigma_s})
\mathrm{vimo}_g^t({}^{z_{s},\dots,z_{s+t}}_{\sigma_{s+1},\dots,\sigma_{s+t}})
-
\mathrm{vimo}_g^t({}^{z_0,\dots,z_t}_{\sigma_1,\dots,\sigma_t})
\mathrm{vimo}_f^s({}^{z_{t},\dots,z_{s+t}}_{\sigma_{t+1},\dots,\sigma_{s+t}}).
$$
While by our induction assumption we have
\begin{align*}
&\mathrm{vimo}_h^r({}^{0,z_1-z_0,\dots,z_r-z_0}_{\qquad \sigma_1,\dots,\sigma_r}) \\
&=\mathrm{vimo}_f^s({}^{0,z_1-z_0,\dots,z_s-z_0}_{\qquad\sigma_1,\dots,\sigma_s})
\mathrm{vimo}_g^t({}^{z_{s}-z_0,\dots,z_{s+t}-z_0}_{\qquad\sigma_{s+1},\dots,\sigma_{s+t}}) \\
&\qquad\qquad \qquad \qquad \qquad  \qquad \qquad
-
\mathrm{vimo}_g^t({}^{0,z_1-z_0,\dots,z_t-z_0}_{\qquad\sigma_1,\dots,\sigma_t})
\mathrm{vimo}_f^s({}^{z_{s}-z_0,\dots,z_{s+t}-z_0}_{\qquad\sigma_{t+1},\dots,\sigma_{s+t}})       \\
&=\mathrm{vimo}_f^s({}^{z_0,\dots,z_s}_{\sigma_1,\dots,\sigma_s})
\mathrm{vimo}_g^t({}^{z_{s},\dots,z_{s+t}}_{\sigma_{s+1},\dots,\sigma_{s+t}})
-
\mathrm{vimo}_g^t({}^{z_0,\dots,z_t}_{\sigma_1,\dots,\sigma_t})
\mathrm{vimo}_f^s({}^{z_{t},\dots,z_{s+t}}_{\sigma_{t+1},\dots,\sigma_{s+t}})  \\
&=\mathrm{vimo}_h^r({}^{z_0,\dots,z_r}_{\sigma_1,\dots,\sigma_r}).
\end{align*}

The equation \eqref{eq:mantar invariance for ma} is a formal generalization of
\cite[Lemma A.2]{S}. We give a short proof below.
Since $h\in\mathbb L_w$, we have
$h=(-1)^{w-1}\mathrm{anti}(h)$.
Thus by definition, we have
$
\mathrm{vimo}_h^r({}^{z_0,\dots,z_r}_{\sigma_1,\dots,\sigma_r})
=(-1)^{w-1}\mathrm{vimo}_h^r({}^{z_r,\dots,z_0}_{\sigma_r,\dots,\sigma_1})
$.
Therefore
\begin{align*}
\mathrm{ma}_h^r&({}^{z_1,\dots,z_r}_{\sigma_1,\dots,\sigma_r})
=\mathrm{vimo}_h^r({}^{0,z_1,z_1+z_2,\dots,z_1+\cdots+z_r}_{\qquad \sigma_1,\dots,\sigma_r})
=(-1)^{w-1}\mathrm{vimo}_h^r({}^{z_1+\cdots+z_r, \dots, z_1+z_2,z_1,0}_{\qquad \sigma_r,\dots,\sigma_1})\\
&=(-1)^{r-1}\mathrm{vimo}_h^r({}^{-z_1-\cdots-z_r, \dots, -z_1-z_2,-z_1,0}_{\qquad \quad\sigma_r,\dots,\sigma_1})
=(-1)^{r-1}\mathrm{vimo}_h^r({}^{0, z_r, z_r+z_{r-1},\dots, z_r+\cdots+z_1}_{\qquad \quad\sigma_r,\dots,\sigma_1}) \\
&=(-1)^{r-1}\mathrm{ma}_h^r({}^{z_r,\dots,z_1}_{\sigma_r,\dots,\sigma_1})
=\mathrm{mantar}\circ\mathrm{ma}_h^r({}^{z_1,\dots,z_r}_{\sigma_1,\dots,\sigma_r}).
\end{align*}
Here in the third equality we use that $\mathrm{vimo}_h^r$ is homogeneous with degree $w-r$
and the fourth equality is by \eqref{eq:translation invariance of vimo}.
\end{proof}

\begin{defn}
Let $\mathfrak{tder}$ be the set of tangential derivation of $\mathbb L$,
the derivation  $D_{\{F_\sigma\}_{\sigma}, G}$ of $\mathbb L$
defined by $x\mapsto [x,G]$ and $y_\sigma\mapsto [y_\sigma,F_\sigma]$ for some $F_\sigma,G\in \mathbb L$.
It forms a Lie algebra by the bracket
\begin{equation}\label{bracket}
[D_{\{F^{(1)}_{\sigma}\},G^{(1)}},D_{\{F_\sigma^{(2)}\},G^{(2)}}]
=D_{\{F^{(1)}_{\sigma}\},G^{(1)}}\circ D_{\{F^{(2)}_{\sigma}\},G^{(2)}}
-D_{\{F^{(2)}_{\sigma}\},G^{(2)}}\circ D_{\{F^{(1)}_{\sigma}\},G^{(1)}}.
\end{equation}
The action of $\Gamma$ on $\mathbb L$ induces the $\Gamma$-action on $\mathfrak{tder}$.
We denote its invariant part by $\mathfrak{tder}^\Gamma$.
We mean $\mathfrak{sder}$ to be set of special derivations,
tangential derivations such that $D_{\{F_{\sigma}\},G}(x+\sum_\sigma y_\sigma)=0$
and $\mathfrak{sder}^\Gamma$ to be
its intersection with $\mathfrak{tder}^\Gamma$,
both of which forms a Lie subalgebra of $\mathfrak{tder}$.
We put
$\mathfrak{mt}:=\oplus_{(w,d)\neq (1,0)}\mathbb L_{w,d}$. It forms a Lie algebra
by the bracket
\begin{equation}\label{eq:mt bracket}
\{f_1,f_2\}=D_{\{\sigma(f_1)\},0}(f_2)-D_{\{\sigma(f_2)\},0}(f_1)+[f_1,f_2],
\end{equation}
in other words, $D_{\{\sigma(\{f_1,f_2\})\},0}:=[D_{\{\sigma(f_1)\},0},D_{\{\sigma(f_2)\},0}]$.
\end{defn}

We occasionally regard  $\mathfrak{mt}$  as a Lie subalgebra of $\mathfrak{tder}$
by $f\mapsto D_{\{\sigma(f)\},0}$.
We note that the condition \eqref{KV1} is equivalent to $D_{\{\sigma(F)\},G}\in\mathfrak{sder}^\Gamma$.

We regard
$\mathfrak{sder}^\Gamma$ and $\mathfrak{mt}$ as filtered  graded Lie algebras
by encoding them with
$\mathrm{Fil}_\mathcal{D}^d\mathfrak{sder}^\Gamma_w=\{D_{\{\sigma(F)\},G}\in\mathfrak{sder}^\Gamma \bigm|
F\in\mathrm{Fil}_\mathcal{D}^d\mathbb L_w \}$ and
$\mathrm{Fil}_\mathcal{D}^d\mathfrak{mt}=\{D_{\{\sigma(F)\},0}\in\mathfrak{mt} \bigm|
F\in\mathrm{Fil}_\mathcal{D}^d\mathbb L_w \}$.
The following is required in the next section.
\begin{lem}
We have  a  natural graded Lie algebra homomorphism
\begin{equation}\label{eq:res}
\mathrm{res}:\mathrm{gr}_\mathcal{D}(\mathfrak{sder}^\Gamma)\to\mathfrak{mt}
\end{equation}
sending $D_{\{\tau(F)\},G}\mapsto D_{\{\tau(\bar F)\},0}$.
\end{lem}

\begin{proof}
Let $D_i=D_{\{\tau(F_i)\},G_i}\in \mathrm{Fil}_\mathcal{D}^{d_i}\mathfrak{sder}^\Gamma_{w_i}$ for $i=1,2$.
Put $D_3=[D_1,D_2]$. Since it belongs to $\mathfrak{sder}^\Gamma$,
$D_3$ is expressed as $D_{\{\tau(F_3)\},G_3}$ for some $F_3\in\mathbb L$
and $G_3\in\mathbb L$.

By \eqref{KV1}, we have  $\tau(F_i)\in\mathrm{Fil}^{d_i}_{\mathcal{D}}\mathbb L_{w_i}$,
$G_i\in\mathrm{Fil}^{d_i+1}_{\mathcal{D}}\mathbb L_{w_i}$ ($i=1,2$).
Since
$F_3$ is calculated to be
$$
F_3=D_{\{\tau(F_1)\},G_1}(F_2)-D_{\{\tau(F_2)\},G_2}(F_1)+[F_1,F_2]
$$
by \eqref{bracket},
we see that $F_3\in\mathrm{Fil}^{d_1+d_2}_{\mathcal{D}}\mathbb L_{w_1+w_2}$.
The residue class $\bar F_3\in \mathrm{gr}^{d_1+d_2}_{\mathcal{D}}\mathbb L_{w_1+w_2}$ is
calculated as
\begin{equation}\label{eq:abuse bracket}
\bar F_3=D_{\{\tau(\bar F_1)\},0}(\bar F_2)-D_{\{\tau(\bar F_2)\},0}(\bar F_1)+[\bar F_1, \bar F_2],
\end{equation}
where  $\bar F_1\in\mathrm{gr}^{d_1}_{\mathcal{D}}\mathbb L_{w_1}$ and $\bar F_2\in\mathrm{gr}^{d_2}_{\mathcal{D}}\mathbb L_{w_2}$ are the residue classes of $F_1$ and $F_2$.
Therefore our map is a Lie algebra homomorphism.
\end{proof}

\begin{prop
\label{prop:MT=ARIal}
The map $\mathrm{ma}$ sending $h\to\mathrm{ma}_h$ induces
a filtered graded Lie algebra isomorphism
\begin{equation}\label{eq:hom ma}
\mathrm{ma}: \mathfrak{mt}\simeq \mathrm{ARI}(\Gamma)^{\mathrm{fin},\mathrm{pol}}_{\mathrm{al}}
\end{equation}
where
$\mathrm{ARI}(\Gamma)^{\mathrm{fin},\mathrm{pol}}_{\mathrm{al}}$ is  the  Lie algebra (cf. Lemma \ref{ARIal Lie algebra})
equipped with the $\mathrm{ari}_u$-bracket.
\end{prop}

\begin{proof}
It can be proved completely in a same way to that of {\cite[Theorem 3.4.2]{S-ARIGARI}}.
\end{proof}

We prepare the following technical lemma which is
required to  the proof of a reformulation of \eqref{KV1}
in Lemma \ref{lem:refo1:KV1}.

\begin{lem}\label{lem:aux H=[x,G]}
Let $H\in\mathbb L_w$ with $w\geqslant 2$.
Assume that $H$ has no words starting with any $y_\sigma$ and ending in any $y_\tau$.
Then  there exists  $G\in\mathbb L_{w-1}$ such that $H=[x,G]$.
\end{lem}

\begin{proof}
The proof goes on the same way to the proof of  \cite[Proposition 2.2]{S}.
Define  the derivation $\partial_x$ of $\mathbb A$ sending $x\mapsto 1$ and
$y_\sigma\mapsto 0$ and the $\mathbb Q$-linear endomorphism $\sec$ of $\mathbb A$ by
$\sec(h):=\sum_{i\geqslant 0}\frac{(-1)^i}{i!}\partial_x^i(h)x^i$
for $h\in\mathbb A$.
Let us write  $H=H_xx+\sum_\sigma H_{y_\sigma}y_\sigma$.
Then by our assumption,  we have $P\in\mathbb A$ such that  $xP=\sum_\sigma H_{y_\sigma}y_\sigma$.
Then by \cite{R} Proposition 4.2.2 and
$\partial_x^i(xP)=i\partial_x^{i-1}(P)+x\partial_x^{i}(P)$, we have
\begin{align*}
H&=\sec(\sum_\sigma H_{y_\sigma}y_\sigma)=\sec(xP)
=\sum_{i\geqslant 0}\frac{(-1)^i}{i!}\partial_x^i(xP)x^i\\
&=\sum_{i\geqslant 1}\frac{(-1)^i}{(i-1)!}\partial_x^{i-1}(P)x^i
+x\sum_{i\geqslant 0}\frac{(-1)^i}{i!}\partial_x^i(P)x^i
=-\sec(P)x+x\sec(P)=[x,\sec(P)].
\end{align*}
It remains to show that $G=\sec(P)$ is in $\mathbb L$,
which follows from exactly the same  arguments  to the last half of
the proof of  \cite[Proposition 2.2]{S}.
\end{proof}

Let $F=F(x;(y_\sigma))\in\mathbb L$. As in \cite{S}, we put
\begin{equation}\label{eq:f and F}
f(x;(y_\sigma))=F(z;(y_\sigma)) \quad \text{ and } \quad\tilde f(x;(y_\sigma))=f(x;(-y_\sigma))
\end{equation}
with $z=-x-\sum_{\sigma}y_\sigma$.

The following generalizes the equivalence between (i) and (v)
in \cite[Theorem 2.1]{S}.

\begin{lem
\label{lem:refo1:KV1}
Let $F\in\mathbb L_w$ with $w\geqslant 1$.
Then saying \eqref{KV1} for $F$ is equivalent to
saying that
\begin{equation}\label{eq:refo1:KV1}
\{\tilde f_{y_{\gamma}}+\tilde f_x\}
=(-1)^{w-1}\gamma\cdot\mathrm{anti}\{\tilde f_{y_{\gamma^{-1}}}+\tilde f_x\}
\end{equation}
for all $\gamma\in\Gamma$.
\end{lem}

\begin{proof}
Firstly, we show that \eqref{KV1} for $F$ is equivalent to
$\alpha(F)_{y_\beta}=\beta(F)^{y_\alpha}$ for any $\alpha, \beta\in \Gamma$.
Set $H=\sum_\sigma[y_\sigma,\sigma(F)]$.
Then  we have
\begin{equation}\label{eqn for H}
H=\sum_{\alpha,\beta}
y_\alpha\left\{
\alpha (F)_{y_\beta} - \beta(F)^{y_\alpha}
\right\}y_\beta
+ \sum_{\alpha}y_\alpha \alpha(F)_xx
- \sum_{\beta}x\beta(F)^xy_\beta.
\end{equation}

Assume \eqref{KV1} for $F$.
Then
\begin{equation}\label{eqn for H 2}
H=Gx-xG.
\end{equation}
So $H$ has no words starting with any $y_\sigma$ and ending in any $y_\tau$.
By \eqref{eqn for H}, we have $\alpha(F)_{y_\beta}=\beta(F)^{y_\alpha}$.
Conversely assume $\alpha(F)_{y_\beta}=\beta(F)^{y_\alpha}$.
Then  $H$ has no words starting with any $y_\sigma$ and ending in any $y_\tau$.
By Lemma \ref{lem:aux H=[x,G]},
there is a $G\in\mathbb L_{w}$
such that $H=[G,x]$.
Whence we get \eqref{KV1}.

Secondly,
by $\alpha(F)\in\mathbb L_w$, we have $\alpha(F)^{y_\beta}=(-1)^{w-1}\mathrm{anti}(\alpha(F)_{y_\beta})$.
Therefore $\alpha(F)_{y_\beta}=\beta(F)^{y_\alpha}$
is equivalent to
$\alpha(F)_{y_\beta}=(-1)^{w-1}\mathrm{anti}(\beta(F)_{y_\alpha})$.

Lastly,
by \eqref{eq:f and F}
we have
$
f_{y_\alpha}(z;(y_\sigma))-f_x(z;(y_\sigma))=F_{y_\alpha}.
$
Hence
\begin{equation*}
\alpha(F)_{y_\beta}=\alpha(F_{y_{\alpha^{-1}\beta}})=
\alpha\left\{f_{y_{\alpha^{-1}\beta}}(z;(y_\sigma))-f_x(z;(y_\sigma))\right\}
\end{equation*}
Whence
$\alpha(F)_{y_\beta}=(-1)^{w-1}\mathrm{anti}(\beta(F)_{y_\alpha})$
is equivalent to
$$
\alpha\left\{f_{y_{\alpha^{-1}\beta}}(z;(y_\sigma))-f_x(z;(y_\sigma))\right\}
=(-1)^{w-1}\mathrm{anti}\left(
\beta\left\{f_{y_{\alpha\beta^{-1}}}(z;(y_\sigma))-f_x(z;(y_\sigma))\right\}
\right)
$$
from which our claim follows.
\end{proof}

The following reformulation is suggested by the arguments in \cite[Appendix A]{S}.

\begin{lem}\label{lem:refo2:KV1}
Let $\tilde f\in\mathbb L_w$ with $w>1$.
Then
\eqref{eq:refo1:KV1} for all $\gamma\in\Gamma$
is equivalent to  the following
{\it senary relation}
\footnote{
It is because it consists of 6 terms, 3 terms on each hand sides.
}
(cf. \cite[(3.64)]{E-flex})
\begin{equation}\label{senary relation}
\mathrm{teru}( M)^r=\mathrm{push}\circ\mathrm{mantar}\circ\mathrm{teru}\circ\mathrm{mantar}( M)^r
\end{equation}
for $1\leqslant r \leqslant w$
with $ M=\mathrm{ma}_{\tilde f}$.

\end{lem}

\begin{proof}

Because $M=\mathrm{ma}_{\tilde f}$ is $\mathrm{mantar}$ invariant for $\tilde f\in\mathbb L_w$
by \eqref{eq:mantar invariance for ma},
the senary relation \eqref{senary relation} is equivalent to
\begin{equation}\label{senary relation of mantar invariant}
\mathrm{swap}\circ\mathrm{teru}( M)^r
=\mathrm{swap}\circ\mathrm{push}\circ\mathrm{mantar}\circ\mathrm{teru}( M)^r.
\end{equation}
For simplicity, we denote $\sigma_i\cdots\sigma_j$ by $\sigma_{i,j}$ for $1\leqslant i\leqslant j\leqslant r$. By definition, its left hand side is calculated to be
\begin{align}\label{LHS of senary of mantar invariant}
	&\mathrm{vimo}^r_{\tilde f}
	{\scriptsize\left(\begin{array}{rrrrr}
		0,& v_r,& \dots,& v_2,& v_1 \\
		& \sigma_{1,r},& \dots,& \sigma_{1,2},& \sigma_1
	\end{array}\right)} \\
	&\hspace{0cm}+\frac{1}{v_1-v_2}\left\{\mathrm{vimo}^{r-1}_{\tilde f}
		{\scriptsize\left(\begin{array}{rrrrr}
			0,& v_r,& \dots,& v_3,& v_1 \\
			& \sigma_{1,r},& \dots,& \sigma_{1,3},& \sigma_{1,2}
		\end{array}\right)}
	-\mathrm{vimo}^{r-1}_{\tilde f}
	{\scriptsize\left(\begin{array}{rrrrr}
		0,& v_r,& \dots,& v_3,& v_2 \\
		& \sigma_{1,r},& \dots,& \sigma_{1,3},& \sigma_{1,2}
	\end{array}\right)}\right\}. \nonumber
\end{align}
By
$\mathrm{vimo}_{\tilde f^r}^r\varia{z_0,\dots,z_r}{\sigma_1,\dots,\sigma_r} =(-1)^{w-r}\mathrm{vimo}_{\tilde f^r}^r\varia{-z_0,\dots,-z_r}{\ \ \sigma_1,\dots,\sigma_r}$
 and definition, its right hand side is calculated to be
\begin{align}\label{RHS of senary of mantar invariant}
	&(-1)^{w-1}\left[\mathrm{vimo}^r_{\tilde f}
	{\scriptsize\left(\begin{array}{rrrrrr}
		0,& v_3-v_2,& \dots,& v_r-v_2,& -v_2,& v_1-v_2 \\
		& \sigma_{2,2},& \dots,& \sigma_{2,r-1},& \sigma_{2,r},& \sigma_1^{-1}
	\end{array}\right)}\right. \\
	&\hspace{0cm}\left.+\frac{1}{v_1}\left\{\mathrm{vimo}^{r-1}_{\tilde f}
		{\scriptsize\left(\begin{array}{rrrrr}
			0,& v_3-v_2,& \dots,& v_r-v_2,& v_1-v_2 \\
			& \sigma_{2,2},& \dots,& \sigma_{2,r-1},& \sigma_{2,r}
		\end{array}\right)}
	-\mathrm{vimo}^{r-1}_{\tilde f}
	{\scriptsize\left(\begin{array}{rrrrr}
		0,& v_3-v_2,& \dots,& v_r-v_2,& -v_2 \\
		& \sigma_{2,2},& \dots,& \sigma_{2,r-1},& \sigma_{2,r}
	\end{array}\right)}\right\}\right]. \nonumber
\end{align}
By $\mathrm{vimo}_{\tilde f^r}^r\varia{z_0,\dots,z_r}{\sigma_1,\dots,\sigma_r} =\mathrm{vimo}_{\tilde f^r}^r\varia{0,z_1-z_0,\dots,z_r-z_0}{\qquad\sigma_1,\dots,\sigma_r}$
in \eqref{eq:translation invariance of vimo}
and the change of variables
$z_0=-v_1$, $z_1=v_r-v_1$, $\dots$, $z_{r-1}=v_2-v_1$ and $\gamma=\sigma_1$, $\tau_1=\sigma_{1,r}$, $\dots$, $\tau_{r-1}=\sigma_{1,2}$, the formula \eqref{LHS of senary of mantar invariant} is equal to
\begin{align}\label{LHS of senary of mantar invariant-2}
	\mathrm{vimo}^{r}_{{\tilde f}^{r}}\varia{z_0,\dots,z_{r-1},0}{\tau_1,\dots,\tau_{r-1},\gamma}
	+\frac{1}{z_{r-1}}\left\{
		\mathrm{vimo}^{r-1}_{\tilde f^{r-1}}\varia{z_0,\dots,z_{r-1}}{\tau_1,\dots,\tau_{r-1}}
		-\mathrm{vimo}^{r-1}_{\tilde f^{r-1}}\varia{z_0,\dots,z_{r-2},0}{\tau_1,\dots,\tau_{r-2},\tau_{r-1}}
	\right\},
\end{align}
and \eqref{RHS of senary of mantar invariant} is equal to
\begin{align}\label{RHS of senary of mantar invariant-2}
	&(-1)^{w-1}\left[
	\mathrm{vimo}^{r}_{{\tilde f}^{r}}\varia{z_{r-1},\dots,z_0,0}{\gamma^{-1}\tau_{r-1},\dots,\gamma^{-1}\tau_1,\gamma^{-1}} \right. \\
	&\hspace{1cm}\left.+\frac{1}{z_0}\left\{
		\mathrm{vimo}^{r-1}_{\tilde f^{r-1}}\varia{z_{r-1},\dots,z_1,z_0}{\gamma^{-1}\tau_{r-1},\dots,\gamma^{-1}\tau_1}
		-\mathrm{vimo}^{r-1}_{\tilde f^{r-1}}\varia{z_{r-1},\dots,z_1,0}{\gamma^{-1}\tau_{r-1},\dots,\gamma^{-1}\tau_1}
	\right\}
	\right]. \nonumber
\end{align}
Whence \eqref{senary relation} is equivalent to \eqref{LHS of senary of mantar invariant-2}=\eqref{RHS of senary of mantar invariant-2}
for $1\leqslant r \leqslant w$ and $\tau_1,\dots,\tau_{r-1},\gamma\in\Gamma$.

On the other hand, \eqref{eq:refo1:KV1} for all $\gamma\in\Gamma$
is equivalent to
\footnote{
Here ${\tilde f_x^r}$ and ${\tilde f_y^r}$ mean $({\tilde f_x})^r$
and $({\tilde f_y})^r$.
}
\begin{align}\label{eq:refo1:KV1-ver:2}
	&\mathrm{vimo}^r_{\tilde f_{y_{\gamma}}^r}\varia{z_0,\dots,z_r}{\tau_1,\dots,\tau_r}
		+\mathrm{vimo}^r_{\tilde f_x^r}\varia{z_0,\dots,z_r}{\tau_1,\dots,\tau_r} \\
	&=(-1)^{w-1}\left[\mathrm{vimo}^r_{\gamma\cdot\mathrm{anti}(\tilde f_{y_{\gamma^{-1}}}^r)}
		\varia{z_0,\dots,z_r}{\tau_1,\dots,\tau_r}
		+\mathrm{vimo}^r_{\gamma\cdot\mathrm{anti}(\tilde f_x^r)}
		\varia{z_0,\dots,z_r}{\tau_1,\dots,\tau_r}\right] \nonumber
\end{align}
for $\gamma\in\Gamma$ and $1\leqslant r \leqslant w$.
By definition, we have
\begin{align*}
	\mathrm{vimo}^{r+1}_{(\tilde f_{y_{\gamma}})^ry_{\gamma}}\varia{z_0,\dots,z_{r+1}}{\tau_1,\dots,\tau_{r+1}}
	=\delta_{\tau_{r+1},\gamma}\mathrm{vimo}^{r+1}_{{\tilde f}^{r+1}}\varia{z_0,\dots,z_r,0}{\tau_1,\dots,\tau_r,\gamma}
\end{align*}
where $\delta_{\tau_{r+1},\gamma}
:=\left\{\begin{array}{ll}
	1 & (\tau_{r+1}=\gamma), \\
	0 & (\mbox{otherwise}).
\end{array}\right.$
So by using this, we get
\begin{align}\label{vimo y formula}
	\mathrm{vimo}^r_{\tilde f_{y_{\gamma}}^r}\varia{z_0,\dots,z_{r}}{\tau_1,\dots,\tau_{r}}
	=\mathrm{vimo}^{r+1}_{(\tilde f_{y_{\gamma}})^ry_{\gamma}}\varia{z_0,\dots,z_r,0}{\tau_1,\dots,\tau_r,\gamma}
	=\mathrm{vimo}^{r+1}_{{\tilde f}^{r+1}}\varia{z_0,\dots,z_r,0}{\tau_1,\dots,\tau_r,\gamma}.
\end{align}
By ${\tilde f}^r={(\tilde f_{x})}^rx+\sum_{\gamma\in\Gamma}{(\tilde f_{y_{\gamma}})}^{r-1}y_{\gamma}$ and $\mathrm{vimo}^r_{(\tilde f_x)^rx}\varia{z_0,\dots,z_r}{\tau_1,\dots,\tau_r}=z_r\cdot\mathrm{vimo}^r_{\tilde f_x^r}\varia{z_0,\dots,z_r}{\tau_1,\dots,\tau_r}$, we have
\begin{align}\label{vimo x formula}
	\mathrm{vimo}^r_{\tilde f_x^r}\varia{z_0,\dots,z_r}{\tau_1,\dots,\tau_r}
	=\frac{1}{z_r}\left\{
		\mathrm{vimo}^r_{\tilde f^r}\varia{z_0,\dots,z_r}{\tau_1,\dots,\tau_r}
		-\mathrm{vimo}^r_{\tilde f^r}\varia{z_0,\dots,z_{r-1},0}{\tau_1,\dots,\tau_{r-1},\tau_r}
	\right\}.
\end{align}
Hence, by \eqref{vimo y formula} and \eqref{vimo x formula}, the left hand side of \eqref{eq:refo1:KV1-ver:2} is calculated to be
\begin{align}\label{LHS of eq:refo1:KV1-ver:2}
	\mathrm{vimo}^{r+1}_{{\tilde f}^{r+1}}\varia{z_0,\dots,z_r,0}{\tau_1,\dots,\tau_r,\gamma}
	+\frac{1}{z_r}\left\{
		\mathrm{vimo}^r_{\tilde f^r}\varia{z_0,\dots,z_r}{\tau_1,\dots,\tau_r}
		-\mathrm{vimo}^r_{\tilde f^r}\varia{z_0,\dots,z_{r-1},0}{\tau_1,\dots,\tau_{r-1},\tau_r}
	\right\},
\end{align}
and the left hand side of \eqref{eq:refo1:KV1-ver:2} is calculated to be
\begin{align}\label{RHS of eq:refo1:KV1-ver:2}
	&(-1)^{w-1}\left[
	\mathrm{vimo}^{r+1}_{{\tilde f}^{r+1}}\varia{z_r,\dots,z_0,0}{\gamma^{-1}\tau_r,\dots,\gamma^{-1}\tau_1,\gamma^{-1}} \right. \\
	&\hspace{1cm}\left.+\frac{1}{z_0}\left\{
		\mathrm{vimo}^r_{\tilde f^r}\varia{z_r,\dots,z_1,z_0}{\gamma^{-1}\tau_r,\dots,\gamma^{-1}\tau_1}
		-\mathrm{vimo}^r_{\tilde f^r}\varia{z_r,\dots,z_1,0}{\gamma^{-1}\tau_r,\dots,\gamma^{-1}\tau_1}
	\right\}
	\right]. \nonumber
\end{align}
So \eqref{eq:refo1:KV1} for all $\gamma\in\Gamma$
is equivalent to \eqref{LHS of eq:refo1:KV1-ver:2}=\eqref{RHS of eq:refo1:KV1-ver:2} for $1\leqslant r+1 \leqslant w$ and $\tau_1,\dots,\tau_{r},\gamma\in\Gamma$. This is nothing but \eqref{LHS of senary of mantar invariant-2}=\eqref{RHS of senary of mantar invariant-2} for $r-1$ instead of $r$.
Therefore we get the equivalence between \eqref{eq:refo1:KV1}
and
\eqref{senary relation}.
\end{proof}

Next we consider the condition \eqref{KV2}.

\begin{lem}\label{lem:refo10:kv2}
Let $F\in\mathbb L_w$.
Then \eqref{KV2} for $F$ is equivalent to
\begin{equation}\label{eq:refo10:kv2}
\sum_{i\in\mathbb{Z}/r\mathbb{Z}}
a(\mathbf{q}\circ\pi_Y(f):{}^{e_{i},e_{i+1},\dots,e_{i+r-1},0}_{\sigma_{i+1},\sigma_{i+2},\dots,\sigma_{i+r}})=0
\end{equation}
for each $1\leqslant r\leqslant w$, $(\sigma_1,\dots,\sigma_r)\in\Gamma^r$ and
$(e_0,\dots,e_r)\in\mathcal{E}_w^r$
with $f$ as in \eqref{eq:f and F}.
\end{lem}

\begin{proof}
It is immediate that \eqref{KV2} for $F$ is equivalent to
\begin{equation}\label{reformualtion-quasi-push-inv}
\mathrm{tr}(\mathbf{q}\circ\pi_Y(f))=0.
\end{equation}

For $W=u_1\cdots u_w\in\mathbb A_w$ with $u_i=x$ or $y_\sigma$ ($\sigma\in\Gamma$),
we define  its cyclic permutation by $\mathtt{cy}(W):=u_2\cdots u_wu_1\in\mathbb A_w$.
We put $c(W)$ to be the number of its cycles. It divides $w$.
Then the natural pairing in $\mathrm{Cyc}(\mathbb A)$ is calculated by the one in $\mathbb A$
as follows:
When $W=x^{e_0}y_{\sigma_1}\cdots x^{e_{r-1}}y_{\sigma_r}$,
we have
\begin{align*}
\langle \mathrm{tr} &\{
\mathbf{q}\circ\pi_Y(f)
\Bigm|\mathrm{tr} (W)\rangle_{\mathrm{Cyc}(\mathbb A)}
=\bigl\langle
\mathbf{q}\circ\pi_Y(f)
\Bigm|
\sum_{i=0}^{c(W)-1}\mathtt{cy}^i(W)
\bigr\rangle_{\mathbb A} \\
&=\frac{c(W)}{w}\bigl\langle
\mathbf{q}\circ\pi_Y(f)
\bigm|
\sum_{i=0}^{w-1}\mathtt{cy}^i(W)
\bigr\rangle_{\mathbb A}
=\frac{c(W)}{w}\sum_{i\in\mathbb{Z}/r\mathbb{Z}}a(\mathbf{q}\circ\pi_Y(f)
:{}^{e_{i},\dots,e_{i+r-1},0}_{\sigma_{i+1},\dots,\sigma_{i+r}}).
\end{align*}
So we get the claim.
\end{proof}

\begin{lem}\label{lem:refo2:KV2}
Let $f\in\mathbb L_w$.
Then \eqref{eq:refo10:kv2} for $f$ is equivalent to
the pus-neutrality \eqref{pus-neutral} for $\mathrm{swap}(\mathrm{ma}_{\tilde f})$, i.e.
\begin{equation}\label{reform:KV2}
\sum_{i\in\mathbb{Z}/r\mathbb{Z}
\mathrm{pus}^i\circ\mathrm{swap}( M)^r
\varia{\sigma_{1},\dots,\sigma_{r}}{v_{1},\dots,v_{r}}
=0
\end{equation}
with $ M=\mathrm{ma}_{\tilde f}$
for all $1\leqslant r\leqslant w$
and
$\sigma_i\in\Gamma$ ($1\leqslant i\leqslant r$).
\end{lem}

\begin{proof}
We decompose $\tilde f=\sum_r \tilde f^r$ and describe
$\tilde f^r\in  \mathbb L_{w,r}$ as in \eqref{eq: word expansion}.
Then
\begin{equation*}
\mathrm{ma}^r_{\tilde f}({}^{u_1,\dots,u_r}_{\sigma_1,\dots,\sigma_r})
=\underset{e_0=0}{\sum_{(e_0,\dots,e_r)\in\mathcal{E}_w^r}}
a(\tilde f:{}_{\sigma_1^{-1},\dots,\sigma_r^{-1}}^{e_0,\dots,e_r})
u_1^{e_{1}}(u_1+u_2)^{e_{2}}\cdots (u_1+\cdots+ u_r)^{e_{r}}.
\end{equation*}
By $\tilde f^r\in\mathbb L_{w,r}$, we have
$$
a(\tilde f:{}^{e_0,\dots,e_{r}}_{\sigma_1,\dots,\sigma_r})
=(-1)^{w+1} a(\tilde f:{}^{e_r,\dots,e_{0}}_{\sigma_r,\dots,\sigma_1})
$$
because $\mathrm{anti}(\tilde f^r)=(-1)^{w+1}\tilde f^r$. So
\begin{align*}
\mathrm{swap}\circ \mathrm{ma}^r_{\tilde f}({}_{v_1,\dots,v_r}^{\sigma_1,\dots,\sigma_r})
&=\underset{e_0=0}{\sum_{(e_0,\dots,e_r)\in\mathcal{E}_w^r}}
a(\tilde f:{}_{(\sigma_1\cdots\sigma_r)^{-1},\dots,(\sigma_1\sigma_2)^{-1},\sigma_1^{-1}}^{\qquad e_0,e_1,\dots,e_r})
v_r^{e_{1}}v_{r-1}^{e_{2}}\cdots v_1^{e_{r}} \\
&=(-1)^{w+1}\underset{e_0=0}{\sum_{(e_0,\dots,e_r)\in\mathcal{E}_w^r}}
a(\tilde f:{}_{\sigma_1^{-1}, (\sigma_1\sigma_2)^{-1},\dots,(\sigma_1\cdots\sigma_r)^{-1}}^{\qquad e_r,\dots,e_1,e_0})
v_1^{e_{r}}v_2^{e_{r-1}}\cdots v_r^{e_{1}} \\
&=(-1)^{w+1}\underset{e_r=0}{\sum_{(e_0,\dots,e_r)\in\mathcal{E}_w^r}}
a(\tilde f:{}_{\sigma_1^{-1}, (\sigma_1\sigma_2)^{-1},\dots,(\sigma_1\cdots\sigma_r)^{-1}}^{\qquad\quad e_0,\dots,e_{r-1},e_r})
v_1^{e_{0}}v_2^{e_{1}}\cdots v_r^{e_{r-1}} \\
&=(-1)^{w+r+1}\underset{e_r=0}{\sum_{(e_0,\dots,e_r)\in\mathcal{E}_w^r}}
a(f:{}_{\sigma_1^{-1}, (\sigma_1\sigma_2)^{-1},\dots,(\sigma_1\cdots\sigma_r)^{-1}}^{\qquad e_0,\dots,e_{r-1},e_r})
v_1^{e_{0}}v_2^{e_{1}}\cdots v_r^{e_{r-1}} \\
&=(-1)^{w+r+1}\underset{e_r=0}{\sum_{(e_0,\dots,e_r)\in\mathcal{E}_w^r}}
a(\mathbf{q}\circ \pi_Y(f):{}_{\sigma_1^{-1}, \sigma_2^{-1},\dots,\sigma_r^{-1}}^{e_0,\dots,e_{r-1},0})
v_1^{e_{0}}v_2^{e_{1}}\cdots v_r^{e_{r-1}}.
\end{align*}
Here in the last equality we use
$$
a(\mathbf{q}\circ\pi_Y(h):{}_{\quad \tau_1,\dots,\tau_r}^{e_0,\dots,e_{r-1},0})
=a(h:{}_{\tau_1,\tau_1\tau_2,\dots,\tau_1\cdots\tau_r}^{\ e_0,\dots,e_{r-1},0})
$$
for any $h\in\pi ({\mathbb A})$.
Therefore we obtain
\begin{align*}
\sum_{i\in\mathbb{Z}/r\mathbb{Z}}\mathrm{pus}^i & \circ\mathrm{swap}\circ\mathrm{ma}^r_{\tilde f}({}_{v_1,\dots,v_r}^{\sigma_1,\dots,\sigma_r}) \\&
=(-1)^{w+r+1}\sum_{i\in\mathbb{Z}/r\mathbb{Z}}\underset{e_r=0}{\sum_{(e_0,\dots,e_r)\in\mathcal{E}_w^r}}
a(\mathbf{q}\circ \pi_Y(f):{}_{\sigma_{i+1}^{-1},\dots,
\sigma_{i+r}^{-1}}^{ e_i,\dots,e_{i+r-1}, \ 0})
v_1^{e_{0}}v_2^{e_{1}}\cdots v_r^{e_{r-1}}.
\end{align*}
It is immediate to see that it is equivalent to \eqref{eq:refo10:kv2}.
\end{proof}

The following definition of
the mould version of $\mathfrak{krv}_\bullet$ is suggested by our previous lemmas.

\begin{defn}\label{def:ARI-pspushpusnu}
We define the $\mathbb Q$-linear space
$\mathrm{ARI}(\Gamma)_{\mathrm{sena}/\mathrm{pusnu}}$
to be the subset of moulds $ M$ in $\mathrm{ARI}$
which satisfy the senary relation \eqref{senary relation} and
whose $\mathrm{swap}$ satisfy the pus-neutrality
\eqref{reform:KV2}.
\end{defn}

The $\mathbb Q$-linear space $\mathrm{ARI}(\Gamma)_{\mathrm{sena}/\mathrm{pusnu}}$
is filtered by the depth filtration $\{\mathrm{Fil}^m_\mathcal{D}\}_m$.
We note that $\mathrm{Fil}^2_\mathcal{D}\mathrm{ARI}(\Gamma)_{\mathrm{sena}/\mathrm{pusnu}}\cap \mathrm{ARI}(\Gamma)_\mathrm{al}^{\mathrm{fin},\mathrm{pol}}$
is identified with the depth >1-part of
$\mathrm{ARI}^\mathrm{pol}_{\mathrm{al}+\mathrm{sen*circconst}}$
in \cite[Proposition 28]{RS}
when $\Gamma=\{e\}$.
It is because their formula (79), actually (81), agrees with
\eqref{senary relation} for a mould in $\mathrm{ARI}(\Gamma)_\mathrm{al}^{\mathrm{fin},\mathrm{pol}}$
by Lemma \ref{lem:Embd:1}.

There is an embedding of $\mathbb Q$-linear space
\begin{equation}\label{eq:inclusion grARI}
\mathrm{gr}_{\mathcal{D}}\mathrm{ARI}(\Gamma)_{\mathrm{sena}/\mathrm{pusnu}}\hookrightarrow
\mathrm{ARI}(\Gamma)_{\mathrm{push}/\mathrm{pusnu}}
\end{equation}
that is, the associated graded quotient $\mathrm{gr}_{\mathcal{D}}\mathrm{ARI}(\Gamma)_{\mathrm{sena}/\mathrm{pusnu}}$
of the filtered $\mathbb Q$-linear space
$\mathrm{ARI}(\Gamma)_{\mathrm{sena}/\mathrm{pusnu}}$
is embedded to $\mathrm{ARI}(\Gamma)_{\mathrm{push}/\mathrm{pusnu}}$ introduced in Definition \ref{def:ARIpushpusnu}
because
the push-invariance \eqref{push-invariant} is associated graded with
\eqref{senary relation}
by $\mathrm{gr}_{\mathcal{D}}(\mathrm{teru})=\mathrm{id}$ and $\mathrm{mantar}\circ\mathrm{mantar}=\mathrm{id}$.

\begin{thm}\label{thm:reform:krv}
The map sending $F\in\mathbb A\mapsto\mathrm{ma}_{\tilde f}\in\mathcal M({\mathcal F};\Gamma)$ induces an isomorphism
of filtered $\mathbb Q$-linear spaces
\begin{equation}\label{eq: KRV=ARI al+sena/pusnu}
\mathfrak{krv}(\Gamma)_\bullet\sime
\mathrm{ARI}(\Gamma)_{\mathrm{sena}/\mathrm{pusnu}}\cap\mathrm{ARI}(\Gamma)_\mathrm{al}^{\mathrm{fin},\mathrm{pol}}.
\end{equation}
\end{thm}

\begin{proof}
The restriction of our map
decomposes as
\begin{equation}\label{eq:krv into ARIal}
F\in\mathfrak{krv}(\Gamma)_\bullet
\mapsto {\tilde f}\in\mathfrak{mt} \mapsto \mathrm{ma}_{\tilde f}\in\mathrm{ARI}(\Gamma)_\mathrm{al}^{\mathrm{fin},\mathrm{pol}}.
\end{equation}
By
Proposition \ref{prop:MT=ARIal},
we see that it gives an isomorphism $\mathfrak{mt}\simeq\mathrm{ARI}(\Gamma)_\mathrm{al}^{\mathrm{fin},\mathrm{pol}}$ as (actually filtered) $\mathbb Q$-linear spaces.
Thus we get the claim by our previous lemmas.
\end{proof}

\begin{rem}
When $\Gamma=\{e\}$, the above isomorphism \eqref{eq: KRV=ARI al+sena/pusnu} can be recovered by the composition
of the isomorphisms (29),  (71) and Proposition 28
\footnote{
It looks that there is a slip on the isomorphism on \cite[Proposition 28]{RS}
because its right hand side is completed {\it by depth} while its left hand side is not.
}
in \cite{RS}.
\end{rem}

The authors are not sure
if the bigger space $\mathrm{ARI}(\Gamma)_{\mathrm{sena}/\mathrm{pusnu}}$ is equipped with a structure of Lie algebra under the $\mathrm{ari}_u$-bracket
or not
although we show that a related space $\mathrm{ARI}(\Gamma)_{\mathrm{push}/\mathrm{pusnu}}$ forms a Lie algebra
in Theorem \ref{thm: ARIpushpusnu Lie}.

\subsection{Kashiwara-Vergne bigraded Lie algebra}\label{sec:Kashiwara-Vergne bigraded Lie algebra}
Based on our arguments in the previous subsection,
we introduce
a $\Gamma$-variant $\mathfrak{lkrv}(\Gamma)_{\bullet\bullet}$
of the Kashiwara-Vergne bigraded Lie algebra $\mathfrak{lkrv}$  (\cite{RS})
in Definition \ref{defn:lrkv}
and give a mould theoretical interpretation
in Theorem \ref{thm:reform:krv:bigrade}.
As a corollary we show that it forms a Lie algebra in Theorem \ref{thm:lkrv Lie alg}.

\begin{defn}\label{defn:lrkv}
Kashiwara-Vergne bigraded Lie algebra is defined to be the bigraded $\mathbb Q$-linear space
$\mathfrak{lkrv}(\Gamma)_{\bullet\bullet}=\oplus_{w>1,d>0}\mathfrak{lkrv}(\Gamma)_{w,d}$,
where $\mathfrak{lkrv}(\Gamma)_{w,d}$ is the $\mathbb Q$-linear space
consisting of $\bar F\in \mathrm{gr}^d_{\mathcal{D}}\mathbb L_w$ whose lift
$F\in\mathrm{Fil}^d_{\mathcal{D}}\mathbb L_w$ satisfies the following relations,
\lq the {leading-terms}' of \eqref{KV1} and \eqref{KV2},
\begin{align}\tag{LKV1}
\label{DKV1}
& [x,G]+\sum_{\tau\in\Gamma}[y_\tau,\tau(F)] \equiv 0 \mod  \mathrm{Fil}_{\mathcal{D}}^{d+2} \mathbb L_{w+1},\\
\tag{LKV2}
\label{DKV2}
& \mathrm{tr}\circ\mathbf{q}\circ\pi_Y(F)
\equiv 0 \mod  \mathrm{tr}(\mathrm{Fil}_{\mathcal{D}}^{d+1} \mathbb A_{w})
\end{align}
for a certain $G=G(F)\in\mathbb L_w$.
\end{defn}
We note that such $G\in\mathbb L_{w}$ is in $\mathrm{Fil}_{\mathcal{D}}^{d+1} \mathbb L_{w}$ and
is uniquely determined modulo $\mathrm{Fil}_{\mathcal{D}}^{d+2} \mathbb L_{w}$
by \eqref{DKV1}.
We note that, by  \eqref{DKV2} and $\dim\mathbb L_{w,1}=1$, we have
 $$\mathfrak{lkrv}_{w,d}=\{0\} \qquad  \text{ for  } d=1.$$

\begin{rem}
When $\Gamma=\{e\}$, our definition of $\mathfrak{lkrv}(\Gamma)_{\bullet\bullet}$  does not agree with
that of $\mathfrak{lkv}$ in \cite[Definition 5 and 10]{RS},
which is constructed as if a Lie polynomial version of $\mathrm{ARI}(\Gamma)_{\mathrm{push}/\mathrm{pusnu}}$.
However their depth>1-parts  are eventually isomorphic
via Theorem \ref{thm:reform:krv:bigrade}.
\end{rem}

By definition, there is an inclusion of bigraded $\mathbb Q$-linear spaces
\begin{equation}\label{eq:inclusion grKRV}
\mathrm{gr}_{\mathcal{D}}\mathfrak{krv}(\Gamma)_\bullet \hookrightarrow
\mathfrak{lkrv}(\Gamma)_{\bullet\bullet},
\end{equation}
which generalizes \cite[Proposition 2]{RS},
that is,
the associated graded $\mathbb Q$-linear space $\mathrm{gr}_{\mathcal{D}}\mathfrak{krv}(\Gamma)_\bullet$
of the filtered Lie algebra $\mathfrak{krv}(\Gamma)_\bullet$
is embedded to $\mathfrak{lkrv}(\Gamma)_{\bullet\bullet}$.
We do not know if it is an isomorphism.

For $F\in\mathrm{Fil}^d_{\mathcal{D}}\mathbb L_w$,
we put $\mathsf{f}$ to be the element in $\mathbb L_{w,d}$ corresponding to
$(-1)^{w-d}\bar F\in\mathrm{gr}^m_{\mathcal{D}}\mathbb L_w$ under the natural identification
$\mathrm{gr}^d_{\mathcal{D}}\mathbb L_w\simeq \mathbb L_{w,d}$.
By abuse of notation,
\begin{equation}\label{eq:new f}
\mathsf{f}=(-1)^{w-d}\bar F.
\end{equation}
We write $\mathsf{f}=\mathsf{f}_xx+\sum_\sigma\mathsf{f}_{y_\sigma}y_\sigma$.
We also put
$$\tilde \mathsf{f}=\mathsf{f}(x,(-y_\sigma)_\sigma)$$
and write $\tilde\mathsf{f}=\tilde\mathsf{f}_xx+\sum_\sigma\tilde\mathsf{f}_{y_\sigma}y_\sigma$.

The following is a bigraded variant of Lemma \ref{lem:refo1:KV1}.

\begin{lem}\label{lem:refo1:DKV1}
Let  $F\in\mathrm{Fil}^d_{\mathcal{D}}\mathbb L_w$
for $w\geqslant 1$.
Then \eqref{DKV1} for $F$ is equivalent to
\begin{equation}\label{eq::refo1:LKV1}
\tilde\mathsf{f}_{y_\gamma}= (-1)^{w-1}\gamma(\mathrm{anti}(\tilde\mathsf{f}_{y_{{\gamma}^{-1}}}))
\end{equation}
for any
$\gamma\in\Gamma$.
\end{lem}

\begin{proof}
The proof goes  similarly to  Lemma \ref{lem:refo1:KV1}.

Firstly, we show that \eqref{DKV1} for $F$ is equivalent to
$\alpha(F)_{y_\beta}\equiv \beta(F)^{y_\alpha} \mod \mathrm{Fil}_{\mathcal{D}}^{d}\mathbb L_{w-1}$.
Set $H=\sum_\sigma [y_\sigma,\sigma(F)]\in  \mathrm{Fil}_{\mathcal{D}}^{d+1}\mathbb L_{w+1}$.

Assume \eqref{DKV1} for $F$.
Then $H\equiv Gx-xG\mod \mathrm{Fil}_{\mathcal{D}}^{d+2}\mathbb L_{w+1}$.
So the depth $(d+1)$-part of $H$ has no words starting and ending in any  $y_\sigma$.
By \eqref{eqn for H}, we have
$\alpha(F)_{y_\beta}\equiv \beta(F)^{y_\alpha}
\mod \mathrm{Fil}_{\mathcal{D}}^{d}\mathbb L_{w-1}$.

Conversely assume $\alpha(F)_{y_\beta}\equiv \beta(F)^{y_\alpha}
\mod \mathrm{Fil}_{\mathcal{D}}^{d}\mathbb L_{w-1}$.
Then the depth $(d+1)$-part of $H$ has no words starting and ending in any $y_\sigma$.
By Lemma \ref{lem:aux H=[x,G]}, there is a $G\in\mathbb L_{w}$
which express this part as $[G,x]$, i.e.
$H \equiv [G,x]\mod \mathrm{Fil}_{\mathcal{D}}^{d+2}\mathbb L_{w+1}$.
Whence we get \eqref{DKV1}.

Secondly,
by $\alpha(F)\in\mathbb L$, we have $\alpha(F)^{y_\beta}=(-1)^{w-1}\mathrm{anti}(\alpha (F)_{y_\beta})$.
Therefore $\alpha(F)_{y_\beta}\equiv \beta(F)^{y_\alpha}
\mod \mathrm{Fil}_{\mathcal{D}}^{d}\mathbb L_{w-1}$
is equivalent to
$$\alpha(F)_{y_\beta}\equiv (-1)^{w-1}\mathrm{anti}(\beta(F)_{y_\alpha}) \mod \mathrm{Fil}_{\mathcal{D}}^{d}\mathbb L_{w-1}.$$

Lastly,
by \eqref{eq:new f} and $\alpha(F)_{y_\beta}=\alpha(F_{y_{\alpha^{-1}\beta}})$,
it is equivalent to
$\alpha(\mathsf{f}_{y_{\alpha^{-1}\beta}})= (-1)^{w-1}\mathrm{anti}(\beta(\mathsf{f}_{y_{\alpha\beta^{-1}}}))$,
so for $\tilde \mathsf{f}_y$.
\end{proof}

The following might be regarded as  a bigraded variant of  Lemma \ref{lem:refo2:KV1}.

\begin{lem
\label{lem:refo2:DKV1}
Let $\tilde \mathsf{f}\in\mathbb L_{w,d}$ with $w>1$.
Then \eqref{eq::refo1:LKV1} for any $\gamma\in\Gamma$
is equivalent to $\mathrm{push}$-invariance \eqref{push-invariant}
for $ M=\mathrm{ma}_{\tilde \mathsf{f}}$.
\end{lem}

\begin{proof}
The proof goes similarly to Lemma \ref{lem:refo2:KV1}.
The condition  \eqref{eq::refo1:LKV1} for $\gamma\in\Gamma$
is reformulated to
\begin{equation*}
\mathrm{vimo}^{r}_{\tilde \mathsf{f}_{y_{\gamma}}^{r}}\varia{z_0,\dots,z_{r}}{\tau_1,\dots,\tau_{r}}
=(-1)^{w-1}\mathrm{vimo}^{r}_{\gamma\cdot\mathrm{anti}(\tilde \mathsf{f}_{y_{\gamma^{-1}}}^{r})}
		\varia{z_0,\dots,z_{r}}{\tau_1,\dots,\tau_{r}}
\end{equation*}
for $1\leqslant r \leqslant w$,
which is equivalent to
\begin{equation}\label{eqn:another refo:anti-pal}
\mathrm{vimo}^{r+1}_{\tilde \mathsf{f}^{r+1}}\varia{z_0,\dots,z_{r},0}{\tau_1,\dots,\tau_{r},\gamma}
=(-1)^{w-1}\mathrm{vimo}^{r+1}_{\tilde \mathsf{f}^{r+1}}
		\varia{\qquad z_{r},\dots,z_{0},0}{\gamma^{-1}\tau_{r},\dots,\gamma^{-1}\tau_{1},\gamma^{-1}}
\end{equation}
for $1\leqslant r+1 \leqslant w$.

On the other hand, \eqref{push-invariant} can be shown to be equivalent to
$$
\mathrm{vimo}^{r}_{\tilde \mathsf{f}^{r}}\varia{z_0,\dots,z_{r-1},0}{\tau_1,\dots,\tau_{r-1},\gamma}
=(-1)^{w-1}\mathrm{vimo}^{r}_{\tilde \mathsf{f}^{r}}
		\varia{\qquad z_{r-1},\dots,z_{0},0}{\gamma^{-1}\tau_{r-1},\dots,\gamma^{-1}\tau_{1},\gamma^{-1}}
$$
for $1\leqslant r \leqslant w$,
which turns to be equivalent to
\eqref{eqn:another refo:anti-pal}.

\end{proof}

The following is a bigraded variant of Lemma \ref{lem:refo2:KV2}.

\begin{lem}\label{lem:refo2:DKV2}
Then \eqref{DKV2} for $F\in\mathrm{Fil}^d_{\mathcal{D}}\mathbb L_w$ is equivalent to
the $\mathrm{pus}$-neutrality \eqref{pus-neutral} for $ M=\mathrm{swap}(\mathrm{ma}_{\tilde \mathsf{f}})$,
i.e. \eqref{reform:KV2} for all $r\geqslant 1$,
\end{lem}

\begin{proof}
We note that actually only the terms for $r=d$ contributes in the above equation.
Decompose $\tilde \mathsf{f}$ as in
\eqref{eq: word expansion}.
Then by the arguments in the proof of  Lemma \ref{lem:refo2:KV2},
\eqref{reform:KV2} is equivalent to
$$
\sum_{i\in\mathbb{Z}/r\mathbb{Z}}
a(\mathbf{q}\circ \pi_Y(\mathsf{f}):{}_{\sigma_{i+1}^{-1},\dots,
\sigma_{i+r}^{-1}}^{ e_i,\dots,e_{i+r-1}, \ 0})=0
$$
for all $(\sigma_1,\dots,\sigma_r)\in\Gamma^r$ and
$(e_0,\dots,e_{r-1},0)\in\mathcal{E}_w^r$.
It is nothing but
$$
\mathrm{tr}\circ\mathbf{q}\circ\pi_Y(\mathsf{f})=0,
$$
which is equivalent to \eqref{DKV2}.
\end{proof}

\begin{thm}\label{thm:reform:krv:bigrade}
The map sending $\bar F\in\mathrm{gr}_\mathcal{D}\mathbb L\mapsto\mathrm{ma}_{\tilde\mathsf{f}}\in\mathcal M(\mathcal F;\Gamma)$ induces an isomorphism
of  bigraded $\mathbb Q$-linear spaces
$
\mathfrak{lkrv}(\Gamma)_{\bullet\bullet}\simeq
\mathrm{ARI}(\Gamma)_{\mathrm{push}/ \mathrm{pusnu}}\cap \mathrm{ARI}(\Gamma)_\mathrm{al}^{\mathrm{fin},\mathrm{pol}}.
$$
\end{thm}

\begin{proof}
Our map decomposes as
\begin{equation}\label{eq:gr krv to ARIal}
\bar F\in\mathfrak{lkrv}(\Gamma)_{\bullet\bullet}\mapsto {\tilde \mathsf{f}}\in\mathfrak{mt}
\mapsto \mathrm{ma}_{\tilde\mathsf{f}}\in\mathrm{ARI}(\Gamma)_\mathrm{al}^{\mathrm{fin},\mathrm{pol}}.
\end{equation}
The first map is injective and the second one is isomorphic by
Proposition \ref{prop:MT=ARIal}.
Our claim follows by our previous lemmas.
\end{proof}

As a generalization of \cite[Proposition 1]{RS}, we obtain the following.

\begin{thm}\label{thm:lkrv Lie alg}
The space $\mathfrak{lkrv}(\Gamma)_{\bullet\bullet}$ forms a
bigraded Lie algebra under the bracket \eqref{eq:mt bracket}.
\end{thm}

\begin{proof}
As for the morphism \eqref{eq:gr krv to ARIal},
the second map is Lie algebra homomorphism by Proposition \ref{prop:MT=ARIal}.
It is easy to see that the first map  forms a Lie algebra homomorphism
when we encode $\mathfrak{lkrv}(\Gamma)_{\bullet\bullet}$ with the bracket \eqref{eq:mt bracket}.
Thus our claim follows because it is shown that
$\mathrm{ARI}(\Gamma)_{\mathrm{push}/ \mathrm{pusnu}}$ forms a Lie algebra by Theorem \ref{thm: ARIpushpusnu Lie}
and so $\mathrm{ARI}(\Gamma)_\mathrm{al}$  does by Lemma \ref{ARIal Lie algebra}.
\end{proof}

\begin{rem}
By  \eqref{eq:inclusion grARI}, \eqref{eq:inclusion grKRV}, Theorem \ref{thm:reform:krv} and Theorem \ref{thm:reform:krv:bigrade},
we obtain the following commutative diagram
of Lie algebras.
\begin{equation}\label{CD:krv}
\xymatrix{
\mathrm{gr}_{\mathcal{D}}\mathfrak{krv}(\Gamma)_{\bullet}\ar^{\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\simeq}[r]\ar@{^{(}->}[d]& \mathrm{gr}_{\mathcal{D}}(\mathrm{ARI}(\Gamma)_{\mathrm{sena}/ \mathrm{pusnu}}\cap \mathrm{ARI}(\Gamma)_\mathrm{al}^{\mathrm{fin},\mathrm{pol}})\ar@{^{(}->}[d]\\
\mathfrak{lkrv}(\Gamma)_{\bullet\bullet}
\ar^{\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\simeq}[r]&\mathrm{ARI}(\Gamma)_{\mathrm{push}/ \mathrm{pusnu}}\cap \mathrm{ARI}(\Gamma)_\mathrm{al}^{\mathrm{fin},\mathrm{pol}}
}
\end{equation}
The diagram is commutative because
\eqref{eq:gr krv to ARIal} is associated graded with \eqref{eq:krv into ARIal}.
\end{rem}

Similarly to  Definition \ref{def:ARID}, we impose a distribution relation on
$\mathfrak{lkrv}(\Gamma)$.

\begin{defn}\label{def:LKRVD}
For $N\geqslant 1$ and $\Gamma^N=\{g^N\in\Gamma\ |\  g\in\Gamma\}$,
we consider the map $i_N:\mathfrak{lkrv}(\Gamma)_{\bullet\bullet}\to\mathfrak{lkrv}(\Gamma^N)_{\bullet\bullet}$
which  is  induced by
\begin{equation*}
i_N(x)=x, \qquad
i_N(y_{\tau})=
\begin{cases}
y_{\tau}& \text{ when } \tau\in\Gamma^N, \\
0 & \text{ when } \tau\not\in\Gamma^N
\end{cases}
\end{equation*}
and also the map $m_N:\mathfrak{lkrv}(\Gamma)_{\bullet\bullet}\to\mathfrak{lkrv}(\Gamma^N)_{\bullet\bullet}$
which  is  induced by
\begin{equation*}
m_N(x)=Nx, \qquad
m_N(y_{\sigma})=y_{\sigma^N}
\end{equation*}
for $\sigma\in\Gamma$.
We define the following $\mathbb Q$-linear subspace
$$
\mathfrak{lkrvd}(\Gamma)_{\bullet\bullet}:=\{\bar F\in\mathfrak{lkrv}(\Gamma)_{\bullet\bullet}\bigm | i_N(\bar F)=m_N(\bar F) \text{ for all }
N\geqslant 1\}.
$$
\end{defn}

As a corollary of Theorems \ref{thm:reform:krv:bigrade} and \ref{thm:lkrv Lie alg},
we obtain the following corollary.

\begin{cor}\label{cor:reform:lkrvd}
The space $\mathfrak{lkrvd}(\Gamma)$ forms a bigraded Lie algebra and
we have the following isomorphism of bigraded Lie algebras
$
\mathfrak{lkrvd}(\Gamma)_{\bullet\bullet}\simeq
\mathrm{ARI}(\Gamma)_{\mathrm{push}/ \mathrm{pusnu}}\cap \mathrm{ARID}(\Gamma)_\mathrm{al}^{\mathrm{fin},\mathrm{pol}}.
$$
\end{cor}

\begin{proof}
By definition  we see that both $i_N$ and $m_N$ form Lie algebra homomorphisms
and by Proposition \ref{ARI Lie algebra}
$\mathrm{ARI}(\Gamma)$ forms a Lie algebra.
Therefore our claim follows from
the commutativity of the following diagrams and
Proposition \ref{prop:ARID Lie algebra}:
\begin{equation*}
\xymatrix{
\mathfrak{lkrv}(\Gamma)_{\bullet\bullet}\ar[r]\ar[d]^{i_N}& \mathrm{ARI}(\Gamma)\ar[d]^{i_N}\\
\mathfrak{lkrv}(\Gamma^N)_{\bullet\bullet}\ar[r]&\mathrm{ARI}(\Gamma^N),
}
\qquad
\xymatrix{
\mathfrak{lkrv}(\Gamma)_{\bullet\bullet}\ar[r]\ar[d]^{m_N}& \mathrm{ARI}(\Gamma)\ar[d]^{m_N}\\
\mathfrak{lkrv}(\Gamma^N)_{\bullet\bullet}\ar[r]&\mathrm{ARI}(\Gamma^N).
}
\end{equation*}
\end{proof}

\section{Dihedral Lie algebra}\label{sec:dihedral Lie algebra}
By using mould theoretic interpretations of
the bigraded Lie algebra
${\mathbb D}(\Gamma)_{\bullet\bullet}$ ($\Gamma$: a finite abelian group)
with a dihedral symmetry
and of the Kashiwara-Vergne bigraded Lie algebra
$\mathfrak{lkrv}(\Gamma)_{\bullet\bullet}$,
we realize an embedding $\mathrm{Fil}_{\mathcal{D}}^{2}{\mathbb D}(\Gamma)_{\bullet\bullet}\hookrightarrow\mathfrak{lkrv}(\Gamma)_{\bullet\bullet}$
which extends the result of \cite{RS}.

\subsection{Dihedral bigraded Lie algebra}
We recall the definition of the dihedral bigraded Lie algebra
$D(\Gamma)_{\bullet\bullet}$ introduced by Goncharov \cite{G}
and introduce a related Lie algebra ${\mathbb D}(\Gamma)_{\bullet\bullet}$
which contains $D(\Gamma)_{\bullet\bullet}$
in Definitions \ref{defn: Goncharov dihedral bigraded Lie algebra} and \ref{defn: alal dihedral Lie algebra}.

We call an element $f=f(t_1,\dots,t_{m+1})$ in $\mathbb Q[t_1,\dots,t_{m+1}]$ {\it translation invariant} when
$f(t_1,\dots,t_{m+1})=f(t_1+c,\dots,t_{m+1}+c)$ for any $c\in\mathbb Q$.
We often denote this  by $f(t_1:\dots:t_{m+1})$.
We consider a set of collections
\begin{equation}\label{eq:undertilde Z}
\undertilde Z=\left\{\undertilde Z(g_1, \dots, g_m,g_{m+1} |t_1: \dots :t_{m+1})\right\}_{g_1,\dots,g_m\in\Gamma}
\end{equation}
with
$$g_{m+1}=(g_1\cdots g_m)^{-1}$$
of
translation invariant element  in $\mathbb Q[t_1,\dots,t_{m+1}]$.
For such $\undertilde Z$, we associate
\begin{align}\label{eq:Goncharov swap}
\tilde Z&(g_1:\cdots:g_{m+1}|t_1,\dots,t_{m+1}):= \\
&\undertilde Z(g_1^{-1}g_2,g_2^{-1}g_3,\dots,g_m^{-1}g_{m+1},g_{m+1}^{-1}g_1
|t_1:t_1+t_2: \dots :t_1+\cdots+t_m:0)  \notag
\end{align}
with any $g_1,\dots, g_{m+1}\in\Gamma$ and $t_1+\cdots+t_{m+1}=0$ and also
\begin{align}
\notag
Z(g_1:&\cdots:g_{m+1}|t_1:\dots:t_{m+1}) \\
\label{eq: Z1}
&:=\tilde Z(g_1:\cdots:g_{m+1}|
t_1-t_{m+1},t_2-t_1,\dots ,t_m-t_{m-1}, t_{m+1}-t_m) \\
\label{eq: Z2}
& \ =\undertilde Z(g_1^{-1}g_2,g_2^{-1}g_3,\dots,g_m^{-1}g_{m+1},g_{m+1}^{-1}g_1
|t_1:t_2: \dots :t_m:t_{m+1})
\end{align}
with any $g_1,\dots, g_{m+1}\in\Gamma$ and any $t_1, \dots, t_{m+1}$.

\begin{defn}[\cite{G}]
\label{defn: Goncharov dihedral bigraded Lie algebra}
Set-theoretically
the {\it dihedral bigraded Lie algebra}
means the $\mathbb Q$-linear bigraded space
$$D(\Gamma)_{\bullet\bullet}=\bigoplus_{w,m}D(\Gamma)_{w,m},$$
where the bidegree $(w,m)$-part $D(\Gamma)_{w,m}$ is defined to be
the set $\undertilde Z$ in \eqref{eq:undertilde Z}
of
translation invariant elements
in $\mathbb Q[t_1,\dots,t_{m+1}]$
with total degree $w-m$
satisfying

\noindent
(a). {\it the double shuffle relation}, that is,

(a-i).
{\it the harmonic product}
\begin{equation}\label{harmonic product}
\sum\nolimits_{\sigma\in\Sha_{p,q}}\undertilde Z(
g_{\sigma(1)},\dots,g_{\sigma(m)},g_{m+1}|
t_{\sigma(1)}:\dots:t_{\sigma(m)}:t_{m+1})=0
\end{equation}

(a-ii). {\it the shuffle product}
\begin{equation}\label{shuffle product}
\sum\nolimits_{\sigma\in\Sha_{p,q}}
\tilde Z(g_{\sigma(1)}:\dots:g_{\sigma(m)}:g_{m+1}|
t_{\sigma(1)},\dots,t_{\sigma(m)}, t_{m+1})=0
\end{equation}
for any $p,q\geqslant 1$ with $p+q=m$.

\noindent
(b). {\it the distribution relation} for $N\in\mathbb Z$
such that $|N|$ divides $|\Gamma|$,
\begin{equation}\label{distribution relation}
Z(g_1: \dots :g_{m+1} |t_1:\dots:t_{m+1})
=\frac{1}{|\Gamma_N|}\sum_{h_i^N=g_i}Z(h_1: \dots: h_{m+1} |Nt_1:\dots:Nt_{m+1})
\end{equation}
except that  a constant in $t$ is allowed when $m=1$ and $g_1=g_2$.
Here $|\Gamma_N|$ is the  order of the $N$-torsion subgroup of $\Gamma$
and $\Sha_{p,q}$ is defined by \eqref{shuffle permutation}.

\noindent
(c). Additionally  we put
\begin{equation}\label{eq:additional}
Z(e:e|0:0)=0
\end{equation}
for a technical reason.
\end{defn}

Its Lie algebra structure is explained in \cite{G} \S\S 4-5.
In \cite[Theorem 4.1]{G},
it is shown that
the double shuffle relation implies the dihedral symmetry relations:

\begin{thm}[\cite{G} Theorem 4.1]
\label{ds to dihedral}
The double shuffle relation implies
{\it the dihedral symmetry relations},
which consist of
{\it the cyclic symmetry relation}
\begin{equation*}\label{cyclic symmetry relation}
Z(g_1:g_2:\cdots :g_{m+1}|t_1:t_2:\dots:t_{m+1})=
Z(g_2:\dots:g_{m+1}:g_1|t_2:\dots:t_{m+1}:t_1),
\end{equation*}
{\it the inversion relation} (\lq the distribution relation for $N=-1$')
\begin{equation*}\label{inversion relation}
Z(g_1: \dots:g_{m+1} |t_1: \dots :t_{m+1})
=Z(g_1^{-1}: \dots:g_{m+1}^{-1} |-t_1: \dots :-t_{m+1}).
\end{equation*}
{\it the reflection relation}
\begin{equation*}\label{reflection relation}
Z(g_1:g_2:\cdots :g_{m+1}|t_1:\dots:t_m:t_{m+1})=
(-1)^{m+1}Z(g_{m+1}:\cdots : g_1|-t_m:\dots:-t_1:-t_{m+1}),
\end{equation*}
for $m\geqslant 2$.
\end{thm}

The following reformulation of the above dihedral symmetry relations
is useful in our later arguments.
\begin{rem}
The transformation \eqref{eq: Z2} allows us to rewrite
the above dihedral symmetry relations as follows:

\noindent
{\it the cyclic symmetry relation}
\begin{equation}\label{refo cyclic symmetry relation}
\undertilde Z(g_1,g_2,\dots ,g_{m+1}|t_1:t_2:\dots:t_{m+1})=
\undertilde Z(g_2,\dots,g_{m+1},g_1|t_2:\dots:t_{m+1}:t_1),
\end{equation}
{\it the inversion relation}
\begin{equation}\label{refo inversion relation}
\undertilde Z(g_1,g_2,\dots ,g_{m+1}|t_1:t_2:\dots:t_{m+1})=
\undertilde Z(g_1^{-1},g_2^{-1},\dots,g_{m+1}^{-1}|-t_1:-t_2:\dots:-t_{m+1}),
\end{equation}
{\it the reflection relation}
\begin{equation}\label{refo reflection relation}
\undertilde Z(g_1,\dots, g_m ,g_{m+1}|t_1:\dots:t_m:t_{m+1})=
(-1)^{m+1}\undertilde Z(g_m^{-1},\dots , g_1^{-1},g_{m+1}^{-1}|-t_{m}:\dots:-t_1:-t_{m+1})
\end{equation}
with $g_1 g_2\cdots g_{m+1}=1$.
\end{rem}

\begin{defn}\label{defn: alal dihedral Lie algebra}
Goncharov \cite[\S 4.5 and \S 5.2]{G} also introduced a related Lie algebra
$$D'(\Gamma)_{\bullet\bullet}=\oplus_{w,m}{D'}(\Gamma)_{w,m}$$
which consists of the collections $\undertilde Z$
satisfying the shuffle product \eqref{shuffle product},
the cyclic symmetry relation \eqref{cyclic symmetry relation} and
the additional condition \eqref{eq:additional} (cf. \cite[Proposition 4.6]{G}).
For our purpose, we consider its $\mathbb Q$-linear subspace
$${\mathbb D}(\Gamma)_{\bullet\bullet}=\oplus_{w,m}{\mathbb D}(\Gamma)_{w,m}$$
which consists of elements in $D'(\Gamma)_{\bullet\bullet}$
satisfying the double shuffle relations (a).
\end{defn}

By Theorem \ref{ds to dihedral}, we have
$$
D'(\Gamma)_{\bullet\bullet}\supset
\mathbb D(\Gamma)_{\bullet\bullet}\supset
D(\Gamma)_{\bullet\bullet}.
$$

\subsection{Mould theoretic reformulation}
We explain a reformulation of ${\mathbb D}(\Gamma)_{\bullet\bullet}$ and
$D(\Gamma)_{\bullet\bullet}$ in terms of moulds in
Theorem \ref{thm:reform:dihedral}
and Corollary \ref{cor:reform:dihedral} respectively.

Let $\undertilde Z$ be a collection \eqref{eq:undertilde Z}
of translation invariant elements.
We associate a mould
$${M}_{\undertilde Z}=({M}_{\undertilde Z}^i)_{i\in\mathbb Z_{\geqslant 0}}\in \mathcal M(\mathcal F;\Gamma)$$
by
${M}_{\undertilde Z}^i=0$ for $i\neq m$ and
$${M}_{\undertilde Z}^m({}^{u_1,\dots,u_m}_{g_1,\cdots,g_m})=\tilde Z(g_1:\cdots:g_{m}:1|u_1,\dots,u_{m+1}).$$

The following is a generalization of the results in \cite{M}, \cite{S-ARIGARI}
which treat the case of $\Gamma=\{e\}$.

\begin{thm
\label{thm:reform:dihedral}
The map sending $M:\undertilde Z\in D'(\Gamma)_{\bullet\bullet}
\mapsto M_{\undertilde Z}\in \mathrm{ARI}(\Gamma)$
forms a Lie algebra homomorphism
and  it induces an isomorphism between
\begin{equation}\label{eq:Fil2D=Fil2ARIalal}
\mathrm{Fil}^2_\mathcal{D} {\mathbb D}(\Gamma)_{\bullet\bullet}\simeq
\mathrm{Fil}^2_\mathcal{D} \mathrm{ARI}(\Gamma)_{\underline{\mathrm{al}}/ \underline{\mathrm{al}}}^{\mathrm{fin},\mathrm{pol}}.
\end{equation}
Here the left hand side means the depth>1-part of ${\mathbb  D}(\Gamma)_{\bullet\bullet}$
and the right hand side means the finite polynomial-valued part of
the subset of
$\mathrm{ARI}(\Gamma)_{\underline{\mathrm{al}}/\underline{\mathrm{al}}}$
(cf. Definition \ref{def:ARIalal})
consisting of $M$ with depth>1.
\end{thm}

\begin{proof}
It is immediate that  that \eqref{shuffle product} is equivalent to
the condition for ${M}_{\undertilde Z}$ being in $\mathrm{ARI}(\Gamma)_\mathrm{al}$.
By \eqref{eq:Goncharov swap}, we have
\begin{align*}
\mathrm{swap}(& M^m_{\undertilde Z})\varia{g_1,\dots,g_m}{v_1,\dots,v_m}\\
&=\tilde Z(g_1\cdots g_m:g_1\cdots g_{m-1}:\dots: g_1:1|v_m,v_{m-1}-v_m,\dots,v_2-v_3,v_1-v_2,-v_1)\\
&=\undertilde Z(g_m^{-1},g_{m-1}^{-1},\dots,g_1^{-1},g_1\cdots g_m\ |\ v_m:\dots:v_2:v_1:0).
\end{align*}
Thus we see that \eqref{harmonic product} is equivalent to
the condition for $\mathrm{swap}({M}_{\undertilde Z})$ being in $\overline{\mathrm{ARI}}(\Gamma)_\mathrm{al}$.
Therefore our map forms a $\mathbb Q$-linear isomorphism \eqref{eq:Fil2D=Fil2ARIalal}
by Theorem \ref{ds to dihedral}.

Since $M_{\undertilde Z}$ is in $\mathrm{ARI}(\Gamma)_\mathrm{al}^{\mathrm{fin},\mathrm{pol}}$
when $\undertilde Z\in D'(\Gamma)_{\bullet\bullet}$,
by Proposition \ref{prop:MT=ARIal}
there is an $h\in\mathbb L$ with depth $m$ such that
$$\mathrm{ma}(h)=M_{\undertilde Z},$$
that is,
$$
\mathrm{ma}_h^m({}^{u_1,\dots,u_m}_{g_1,\cdots,g_m})=\tilde Z(g_1:\cdots:g_{m}:1|u_1,\dots,u_{m+1})
$$
with $u_1+\cdots+ u_m+u_{m+1}=0$.
By \eqref{eq:translation invariance of vimo} and \eqref{eq: Z1}, we have
$$
\mathrm{vimo}^m_h({}^{u_0,\dots,u_m}_{g_1,\cdots,g_m})=
Z(g_1:\cdots:g_{m}:1|u_1:\dots :u_{m}:u_0).
$$

In \cite[Theorem 5.2]{G},  $D'(\Gamma)_{\bullet\bullet}$ is realized as  a Lie subalgebra
of $\mathfrak{sder}^\Gamma$ under the morphism
\begin{equation}\label{eq:map xi prime}
\xi'_\Gamma: D'(\Gamma)_{\bullet\bullet}\to \mathrm{gr}_\mathcal{D}\mathfrak{sder}^\Gamma
\end{equation}
sending each
$\undertilde Z=\left\{\undertilde Z(g_1,\dots,g_{m+1}|t_1:\dots:t_{m+1})\right\}_{g_1,\dots,g_{m+1}\in\Gamma}
\in D_{w,m}(\Gamma)$
to the residue class of $D_{\{\sigma(F)\}_\sigma,G(F)}\in\mathfrak{sder}^\Gamma$
in $\mathrm{gr}_\mathcal{D}^m\mathfrak{sder}^\Gamma$
with
\begin{align*}
F&=|\Gamma|^{-1}
\sum_{n_1,\dots,n_{m+1}>0
\atop g_1,\dots,g_{m+1}\in\Gamma}
I_{n_1,\dots,n_{m+1}}(g_1:\cdots:g_{m+1})X^{n_1-1}Y_{g_1^{-1}g_2}\cdots
X^{n_{m}-1}
Y_{g_1^{-1}g_{m+1}}X^{n_{m+1}-1} \\
&=
\sum_{n_0,\dots,n_{m}>0
\atop g_1,\dots,g_{m}\in\Gamma}
I_{n_0,\dots,n_{m}}(1:g_1:\cdots:g_{m})X^{n_0-1}Y_{g_1}\cdots
X^{n_{m-1}-1}
Y_{g_{m}}X^{n_{m}-1}\in\mathbb L_{w,m}
\end{align*}
when the associated element
$Z=\left\{Z(g_1:\cdots:g_{m+1}|t_1:\dots:t_{m+1})\right\}_{g_1,\dots,g_{m+1}\in\Gamma}
$ given by \eqref{eq: Z2} is expressed as
$$
Z(g_1:\cdots:g_{m+1}|t_1:\dots:t_{m+1}):
=\sum_{n_1,\dots,n_{m+1}>0}I_{n_1,\dots,n_{m+1}}(g_1:\cdots: g_{m+1})t_1^{n_1-1}\cdots t_{m+1}^{n_{m+1}-1}.
$$
By combining the Lie algebra homomorphisms $\xi'_\Gamma$ with \eqref{eq:res} and \eqref{eq:hom ma},
we obtain
$$
\mathrm{ma}\circ\mathrm{res}\circ\xi'_\Gamma(\undertilde Z)=\mathrm{ma}(F).
$$

By definition, we have
\begin{align*}
\mathrm{vimo}_F^m({}^{u_1,\dots,u_m}_{g_1^{-1},\dots,g_m^{-1}})
&
\sum_{n_1,\dots,n_{m+1}>0}I_{n_0,\dots,n_{m}}(1:g_1:\cdots:g_{m})
u_0^{n_0-1}\cdots u_m^{n_m-1} \\
&
Z(1:g_1:\cdots:g_m|u_0:\cdots: u_m). \\
\intertext{By the cyclic symmetry relation \eqref{cyclic symmetry relation}, }
&
Z(g_1:\cdots:g_m:1|u_1:\cdots: u_m:u_0)

\mathrm{vimo}_h^m({}^{u_1,\dots,u_m}_{g_1,\dots,g_m}).
\end{align*}
Therefore we have
$
F(x, (y_\sigma)_\sigma)
h(x,(y_{\sigma^{-1}})_\sigma).
$
So
$$M(\undertilde Z)=\mathrm{ma}(h)
\iota\circ\mathrm{ma}\circ\mathrm{res}\circ\xi'_\Gamma(\undertilde Z)$$
where $\iota$ is the map defined by
$(\iota M)^m({}^{u_1,\dots,u_m}_{g_1,\dots,g_m})=
M^m({}^{u_1,\dots,u_m}_{g_1^{-1},\dots,g_m^{-1}})$
which forms a Lie algebra homomorphism.
Since $\iota$, $\mathrm{ma}$ in \eqref{eq:hom ma},
$\mathrm{res}$ in \eqref{eq:res} and
$\xi'_\Gamma$ in \eqref{eq:map xi prime}
are all Lie algebra homomorphisms,
we see that $M$ is so. Whence we get our claims.
\end{proof}

Since
$\mathrm{ARI}(\Gamma)_{\underline{\mathrm{al}}/ \underline{\mathrm{al}}}$,
and hence the right hand side of \eqref{eq:Fil2D=Fil2ARIalal},
forms a Lie algebra by Proposition \ref{ARIalal Lie algebra},
we learn that
$\mathrm{Fil}^2_\mathcal{D} {\mathbb D}(\Gamma)_{\bullet\bullet}$
forms a Lie subalgebra of $D'(\Gamma)_{\bullet\bullet}$.

\begin{cor}\label{cor:reform:dihedral}
The map $M$ in Theorem \ref{thm:reform:dihedral} induces
a Lie algebra isomorphism between
\begin{equation}\label{eq:D=ARIDalal}
\mathrm{Fil}^2_\mathcal{D} {D}(\Gamma)_{\bullet\bullet}\simeq
\mathrm{Fil}^2_\mathcal{D} \mathrm{ARID}(\Gamma)_{\underline{\mathrm{al}}/ \underline{\mathrm{al}}}^{\mathrm{fin},\mathrm{pol}}.
\end{equation}
Here the left hand side means the depth>1-part of ${D}(\Gamma)_{\bullet\bullet}$
and the right hand side means the finite polynomial-valued part of
the subset of
$\mathrm{ARID}(\Gamma)_{\underline{\mathrm{al}}/\underline{\mathrm{al}}}$
(cf. Definition \ref{def:ARID})
consisting of $M$ with depth>1.

\end{cor}

\begin{proof}
Since the distribution relation
\eqref{distribution relation}
is equivalent to
$$
\undertilde Z(g_1, \dots ,g_{m+1} |t_1:\dots:t_{m+1})

\sum_{h_i^N=g_i}\undertilde Z(h_1, \dots, h_{m+1} |Nt_1:\dots:Nt_{m+1}),
$$
which corresponds to
$i_N(M_{\undertilde Z})=m_N(M_{\undertilde Z})$,
that is, $M_{\undertilde Z}\in\mathrm{ARID}(\Gamma)$.
Since
$\mathrm{ARID}(\Gamma)_{\underline{\mathrm{al}}/ \underline{\mathrm{al}}}$,
and hence the right hand side of \eqref{eq:D=ARIDalal},
forms a Lie algebra by
Corollary \ref{cor:ARID alal Lie algebra}, and
$\mathrm{Fil}^2_\mathcal{D} {\mathbb D}(\Gamma)_{\bullet\bullet}$
forms a Lie subalgebra of $D(\Gamma)_{\bullet\bullet}$ by \cite[Theorem 5.2]{G},
our claim follows.
\end{proof}
\subsection{Embedding}
In this subsection, we construct an embedding
$\mathrm{Fil}_{\mathcal{D}}^{2}\mathrm{ARI}(\Gamma)_{\underline{\mathrm{al}}/ \underline{\mathrm{al}}}\hookrightarrow
\mathrm{ARI}(\Gamma)_{\mathrm{push}/ \mathrm{pusnu}}$ in Theorem \ref{thm:embedding}.
As a corollary, we get an embedding
$\mathrm{Fil}_{\mathcal{D}}^{2}{\mathbb D}(\Gamma)_{\bullet\bullet}\hookrightarrow\mathfrak{lkrv}(\Gamma)_{\bullet\bullet}$
in Corollary \ref{cor:embedding}.

The following generalizes {\cite[Lemma 2.5.3]{S-ARIGARI}}.

\begin{lem}
\label{lem:Embd:1}
Any mould $M\in\mathrm{ARI}(\Gamma)_{\mathrm{al}}$ is $\mathrm{mantar}$-invariant
(cf. Notation \ref{nota:mould operations}), namely,
for $m\geqslant1$ and $\sigma_1,\dots,\sigma_m\in\Gamma$, we have
\begin{equation}\label{mantar invariant}
	M^m\varia{x_1,\ \dots,\ x_m}{\sigma_1,\ \dots,\ \sigma_m} = (-1)^{m-1}M^m\varia{x_m,\ \dots,\ x_1}{\sigma_m,\ \dots,\ \sigma_1}.
\end{equation}
\end{lem}
\begin{proof}
For simplicity, we denote $\omega_i:=\binom{x_i}{\sigma_i}$. By using alternality of $M$, we have
\begin{align*}
	\sum_{i=1}^{m-1}(-1)^{i-1}
	\left\{ \sum_{\alpha\in X_{\mathbb{Z}}^\bullet}\Sh{(\omega_i,\dots,\omega_1)}{(\omega_{i+1},\dots,\omega_m)}{\alpha}M^m(\alpha) \right\}
	=0.
\end{align*}
Here, we calculate the left hand side as follows:
\begin{align*}
	\sum_{i=1}^{m-1}&(-1)^{i-1}
		\left\{ \sum_{\alpha\in X_{\mathbb{Z}}^\bullet}\Sh{(\omega_i,\dots,\omega_1)}{(\omega_{i+1},\dots,\omega_m)}{\alpha}M^m(\alpha) \right\} \\
	=&\sum_{i=1}^{m-1}(-1)^{i-1}
		\left\{ \sum_{\alpha\in X_{\mathbb{Z}}^\bullet}\Sh{(\omega_{i-1},\dots,\omega_1)}{(\omega_{i+1},\dots,\omega_m)}{\alpha}M^m(\omega_i,\alpha) \right. \\
	&\left.+ \sum_{\alpha\in X_{\mathbb{Z}}^\bullet}\Sh{(\omega_i,\dots,\omega_1)}{(\omega_{i+2},\dots,\omega_m)}{\alpha}M^m(\omega_{i+1},\alpha) \right\} \\
	=&\sum_{i=1}^{m-1}(-1)^{i-1}
		\sum_{\alpha\in X_{\mathbb{Z}}^\bullet}\Sh{(\omega_{i-1},\dots,\omega_1)}{(\omega_{i+1},\dots,\omega_m)}{\alpha}M^m(\omega_i,\alpha) \\
	&- \sum_{i=2}^{m}(-1)^{i-1}
		\sum_{\alpha\in X_{\mathbb{Z}}^\bullet}\Sh{(\omega_{i-1},\dots,\omega_1)}{(\omega_{i+1},\dots,\omega_m)}{\alpha}M^m(\omega_i,\alpha) \\
	=&M^m(\omega_1,\omega_2,\dots,\omega_m)
		-(-1)^{m-1}
		M^m(\omega_m,\omega_{m-1},\dots,\omega_1).
\end{align*}
So we obtain \eqref{mantar invariant}.
\end{proof}

We define the parallel translation map $\undertilde{t}:\overline{\mathrm{ARI}}(\Gamma)\rightarrow \overline{\mathrm{ARI}}(\Gamma)$ by
\begin{align*}
	\undertilde{t}(M)^m({\bf y}_m):=
	\left\{\begin{array}{ll}
	M^m({\bf y}_m) & (m=0,1), \\
	M^{m-1}{\scriptsize\left(\begin{array}{rrr}
	\sigma_2,& \dots,& \sigma_m \\
	y_2-y_1,& \dots,& y_m-y_1
\end{array}\right)} & (m\geqslant2),
	\end{array}\right.
\end{align*}
for $M\in \overline{\mathrm{ARI}}(\Gamma)$. For our simplicity, we put
$$\undertilde{u}:=\undertilde{t}\circ\mathrm{swap}.$$

\begin{lem}\label{lem:Embd:2}
Let $M\in \mathrm{ARI}(\Gamma)_{\underline{\mathrm{al}}/ \underline{\mathrm{al}}}$ and $m\geqslant2$ and $\sigma_1,\dots,\sigma_m\in\Gamma$ with $\sigma_1\cdots\sigma_m=1$. Then we have
\begin{equation}\label{eqn:G70}
	\undertilde{u}(M)^m\varia{\sigma_1,\ \dots,\ \sigma_m}{x_1,\ \dots,\ x_m}
	= \undertilde{u}(M)^m\varia{\sigma_2^{-1},\ \dots,\ \ \sigma_m^{-1},\ \sigma_1^{-1}}{-x_2,\ \dots,\ -x_m,\ -x_1}.
\end{equation}
\end{lem}

\begin{proof}
We have
\begin{align*}
	\undertilde{u}(M)^m&\varia{\sigma_1,\ \dots,\ \sigma_m}{x_1,\ \dots,\ x_m}  \\
	=&\mathrm{swap}(M)^{m-1}\varia{\ \quad\sigma_2,\qquad\sigma_3,\ \dots,\ \quad \sigma_{m-1},\ \quad \sigma_m}
	{x_2-x_1,\ x_3-x_1,\ \dots,\ x_{m-1}-x_1,\ x_m-x_1} \\
	=&M^{m-1}\varia{x_m-x_1,\ x_{m-1}-x_m,\ \dots,\ x_2-x_3}
	{\sigma_2\cdots\sigma_m,\ \sigma_2\cdots\sigma_{m-1},\ \dots,\quad \sigma_2}.
	\intertext{By using \eqref{mantar invariant}, we get}
	=&(-1)^{m-2}M^{m-1}\varia{x_2-x_3,\ \dots,\ x_{m-1}-x_m,\ x_m-x_1}
	{\ \quad \sigma_2,\ \dots,\ \sigma_2\cdots\sigma_{m-1},\ \sigma_2\cdots\sigma_m} \\
	=&(-1)^{m-2}\mathrm{swap}(M)^{m-1}\varia{\sigma_2\cdots\sigma_m,\quad \sigma_m^{-1},\ \dots,\quad \sigma_3^{-1}}
	{x_2-x_1,\ x_2-x_m,\ \dots,\ x_2-x_3}.
	\intertext{By \eqref{mantar invariant}, $\sigma_1\cdots\sigma_m=1$ and
	$M\in\mathrm{ARI}(\Gamma)_{\underline{\mathrm{al}}/ \underline{\mathrm{al}}}$, we calculate}
	=&\mathrm{swap}(M)^{m-1}\varia{\quad \sigma_3^{-1},\ \dots,\ \quad \sigma_m^{-1},\quad\sigma_1^{-1}}
	{x_2-x_3,\ \dots,\ x_2-x_m,\ x_2-x_1} \\
	=&\undertilde{u}(M)^m\varia{\sigma_2^{-1},\ \sigma_3^{-1},\ \dots,\ \ \sigma_m^{-1},\ \sigma_1^{-1}}
	{-x_2,\ -x_3,\ \dots,\ -x_m,\ -x_1}.
\end{align*}
Therefore, we obtain the claim.
\end{proof}

The following three relations should be called as the cyclic symmetry relation, the inversion relation and the reflection relation respectively (compare them with \eqref{refo cyclic symmetry relation}, \eqref{refo inversion relation} and \eqref{refo reflection relation}
with $m+1$ replaced with $m$).

\begin{lem}\label{lem:Embd:3}
Let $m\geqslant3$ and $M\in\mathrm{ARI}(\Gamma)_{\underline{\mathrm{al}}/ \underline{\mathrm{al}}}$. Then, for $\sigma_1,\dots,\sigma_m\in\Gamma$ with $\sigma_1\cdots\sigma_m=1$, we have the following:
\begin{align}
	\label{eqn:cyc}&\undertilde{u}(M)^m\varia{\sigma_1,\ \sigma_2,\ \dots,\ \sigma_m}{x_1,\ x_2,\ \dots,\ x_m}
	= \undertilde{u}(M)^m\varia{\sigma_2,\ \dots,\ \sigma_m,\ \sigma_1}{x_2,\ \dots,\ x_m,\ x_1}, \\
	\label{eqn:inv}&\undertilde{u}(M)^m\varia{\sigma_1,\ \sigma_2,\ \dots,\ \sigma_m}{x_1,\ x_2,\ \dots,\ x_m}
	 = \undertilde{u}(M)^m\varia{\sigma_1^{-1},\ \sigma_2^{-1},\ \dots,\ \sigma_m^{-1}}{-x_1,\ -x_2,\ \dots,\ -x_m}, \\
	\label{eqn:ref}&
	\undertilde{u}(M)^m\varia{\sigma_1,\ \dots,\ \sigma_{m-1},\ \sigma_m}{x_1,\ \dots,\ x_{m-1},\ x_m}
	= (-1)^{m}\undertilde{u}(M)^{m}\varia{\ \ \sigma_{m-1}^{-1},\ \dots,\ \sigma_{1}^{-1},\ \ \sigma_m^{-1}}{-x_{m-1},\ \dots,\ -x_{1},\ -x_m}.
\end{align}
\end{lem}

\begin{proof}
It is easy to get \eqref{eqn:inv} from \eqref{eqn:G70} and \eqref{eqn:cyc} and to get \eqref{eqn:ref} from \eqref{mantar invariant}, \eqref{eqn:cyc} and \eqref{eqn:inv}.
So it is enough to prove \eqref{eqn:cyc}.

Firstly, we have
\begin{align*}
	&\sum_{\alpha\in Y_{\mathbb{Z}}^\bullet}
	\Sh{\varia{\sigma_1}{x_1}}{\varia{\sigma_2,\ \dots,\ \sigma_m}{x_2,\ \dots,\ x_m}}{\alpha}\undertilde{u}(M)^m(\alpha) \\
	& =\undertilde{u}(M)^m\varia{\sigma_1,\ \sigma_2,\ \dots,\ \sigma_m}{x_1,\ x_2,\ \dots,\ x_m}
	+\sum_{\alpha\in Y_{\mathbb{Z}}^\bullet}
	\Sh{\varia{\sigma_1}{x_1}}{\varia{\sigma_3,\ \dots,\ \sigma_m}{x_3,\ \dots,\ x_m}}{\alpha}\undertilde{u}(M)^m(\varia{\sigma_2}{x_2},\alpha) \\
	& =\undertilde{u}(M)^m\varia{\sigma_1,\ \dots,\ \sigma_m}{x_1,\ \dots,\ x_m} \\
	&\hspace{2cm}+\sum_{\alpha\in Y_{\mathbb{Z}}^\bullet}
	\Sh{\varia{\ \ \ \sigma_1}{x_1-x_2}}{\varia{\ \quad\sigma_3,\ \dots,\qquad \sigma_m}{x_3-x_2,\ \dots,\ x_m-x_2}}{\alpha}\mathrm{swap}(M)^{m-1}(\alpha).
\end{align*}
So by using alternality of $\mathrm{swap}(M)$, we obtain
\begin{equation}\label{first cyclic term}
	\sum_{\alpha\in Y_{\mathbb{Z}}^\bullet}
	\Sh{\varia{\sigma_1}{x_1}}{\varia{\sigma_2,\ \dots,\ \sigma_m}{x_2,\ \dots,\ x_m}}{\alpha}
	\undertilde{u}(M)^m(\alpha)
	=\undertilde{u}(M)^m\varia{\sigma_1,\ \dots,\ \sigma_m}{x_1,\ \dots,\ x_m}.
\end{equation}
Secondly, we have
\begin{align*}
	&\sum_{\alpha\in Y_{\mathbb{Z}}^\bullet}
	\Sh{\varia{\sigma_1}{x_1}}{\varia{\sigma_2,\ \dots,\ \sigma_m}{x_2,\ \dots,\ x_m}}{\alpha}
	\undertilde{u}(M)^m(\alpha) \\
	& =\undertilde{u}(M)^m\varia{\sigma_2,\ \dots,\ \sigma_m,\ \sigma_1}{x_2,\ \dots,\ x_m,\ x_1}
	+\sum_{\alpha\in Y_{\mathbb{Z}}^\bullet}
	\Sh{\varia{\sigma_1}{x_1}}{\varia{\sigma_2,\ \dots,\ \sigma_{m-1}}{x_2,\ \dots,\ x_{m-1}}}{\alpha}\undertilde{u}(M)^m(\alpha,\varia{\sigma_m}{x_m}).
	\intertext{By applying Lemma \ref{lem:Embd:2} to the second term, we get}
	& =\undertilde{u}(M)^m\varia{\sigma_2,\ \dots,\ \sigma_m,\ \sigma_1}{x_2,\ \dots,\ x_m,\ x_1} \\
	&\hspace{2cm}+\sum_{\alpha\in Y_{\mathbb{Z}}^\bullet}
	\Sh{\varia{\sigma_1}{x_1}}{\varia{\sigma_2,\ \dots,\ \sigma_{m-1}}{x_2,\ \dots,\ x_{m-1}}}{\alpha}
	\undertilde{u}(M)^m(\varia{\ \sigma_m^{-1}}{-x_m},\alpha_-) \\
	& =\undertilde{u}(M)^m\varia{\sigma_2,\ \dots,\ \sigma_m,\ \sigma_1}{x_2,\ \dots,\ x_m,\ x_1} \\
	&\hspace{2cm}+\sum_{\alpha\in Y_{\mathbb{Z}}^\bullet}
	\Sh{\varia{\quad\sigma_1}{x_1-x_m}}{\varia{\ \ \quad\sigma_2,\ \dots,\qquad \sigma_{m-1}}{x_2-x_m,\ \dots,\ x_{m-1}-x_m}}{\alpha}\mathrm{swap}(M)^{m-1}(\alpha_-).
\end{align*}
Here, the symbol $\alpha_-$ is $\varia{\tau_1^{-1},\dots,\tau_m^{-1}}{-u_1,\dots,-u_m}$ for $\alpha=\varia{\tau_1,\dots,\tau_m}{u_1,\dots,u_m}$ and for $\binom{\tau_1}{u_1},\dots,\binom{\tau_m}{u_m}\in Y_{\mathbb{Z}}$. So by using alternality of $\mathrm{swap}(M)$, we obtain
\begin{equation}\label{second cyclic term}
	\sum_{\alpha\in Y_{\mathbb{Z}}^\bullet}
	\Sh{\varia{\sigma_1}{x_1}}{\varia{\sigma_2,\ \dots,\ \sigma_m}{x_2,\ \dots,\ x_m}}{\alpha}
	\undertilde{u}(M)^m(\alpha)
	=\undertilde{u}(M)^m\varia{\sigma_2,\ \dots,\ \sigma_m,\ \sigma_1}{x_2,\ \dots,\ x_m,\ x_1}.
\end{equation}
Therefore, by combining \eqref{first cyclic term} and \eqref{second cyclic term}, we obtain \eqref{eqn:cyc}.
\end{proof}

\begin{lem}
For any mould $M\in\mathrm{ARI}(\Gamma)_{\underline\mathrm{al}/\underline\mathrm{al}}$,
	its $\mathrm{swap}(M)$ is {\it neg-invariant} and {\it mantar-invariant}
	(cf. Notation \ref{nota:mould operations})

	namely we have
	\begin{align}
	\label{eqn:inv2}&\mathrm{swap}(M)^m\varia{\sigma_1,\ \dots,\ \sigma_m}{x_1,\ \dots,\ x_m}
	 = \mathrm{swap}(M)^m\varia{\sigma_1^{-1},\ \dots,\ \sigma_m^{-1}}{-x_1,\ \dots,\ -x_m}, \\
	\label{eqn:ref2}&\mathrm{swap}(M)^m\varia{\sigma_1,\ \dots,\ \sigma_m}{x_1,\ \dots,\ x_m}
	 = (-1)^{m-1}\mathrm{swap}(M)^m\varia{\sigma_m^{-1},\ \dots,\ \sigma_1^{-1}}{-x_m,\ \dots,\ -x_1},
\end{align}
for $m\geqslant 0$.
\end{lem}

\begin{proof}
For $m\geqslant 2$, the first equation follows from \eqref{eqn:inv} and
the second one follows from \eqref{eqn:cyc} and \eqref{eqn:ref}.
For $m=0,1$,  they follow from the definition of $\mathrm{ARI}(\Gamma)_{\underline\mathrm{al}/\underline\mathrm{al}}$.
\end{proof}

The following can be also found in \cite[Lemma 2.5.5]{S-ARIGARI}
but we give a different proof below.

\begin{prop}\label{prop:DKV1}
	Any mould $ M\in\mathrm{ARI}(\Gamma)_{\underline\mathrm{al}/\underline\mathrm{al}}$ is push-invariant \eqref{push-invariant}.
\end{prop}

\begin{proof}
	For $m=0,1$, it is obvious because we have $M^1\varia{x_1}{\sigma_1}=M^1\varia{-x_1}{\sigma_1^{-1}}$.
	Assume $m\geqslant2$. By using \eqref{eqn:cyc}, we have
	\begin{equation*}
		\undertilde{u}(M)^{m+1}\varia{\tau_1,\ \tau_2,\ \dots,\ \tau_{m+1}}{y_1,\ y_2,\ \dots,\ y_{m+1}}
		 = \undertilde{u}(M)^{m+1}\varia{\tau_2,\ \dots,\ \tau_{m+1},\ \tau_1}{y_2,\ \dots,\ y_{m+1},\ y_1},
	\end{equation*}
	for $\binom{\tau_1}{y_1},\dots, \binom{\tau_{m+1}}{y_{m+1}}\in Y_{\mathbb{Z}}$ with $\tau_1\cdots\tau_{m+1}=1$. We calculate the left hand side
	\begin{align*}
		\undertilde{u}(M)^{m+1}\varia{\tau_1,\ \tau_2,\ \dots,\ \tau_{m+1}}{y_1,\ y_2,\ \dots,\ y_{m+1}}
		& = \mathrm{swap}(M)^m\varia{\ \quad\tau_2,\ \dots,\ \quad \tau_{m+1}}
		{y_2-y_1,\ \dots,\ y_{m+1}-y_1} \\
		& = M^m
		{\scriptsize\left(\begin{array}{rrrr}
			y_{m+1}-y_1,& y_m-y_{m+1},& \dots,& y_2-y_3 \\
			\tau_2\cdots\tau_{m+1},& \tau_2\cdots\tau_{m},& \dots,& \tau_2
		\end{array}\right)}.
	\end{align*}
	Similarly, we calculate the right hand side
	\begin{align*}
		\undertilde{u}(M)^{m+1}\varia{\tau_2,\ \dots,\ \tau_{m+1},\ \tau_1}{y_2,\ \dots,\ y_{m+1},\ y_1}
		& = M^m
		{\scriptsize\left(\begin{array}{rrrrr}
			y_1-y_2,&  y_{m+1}-y_1,&  y_m-y_{m+1},& \dots,& y_3-y_4 \\
			\tau_3\cdots\tau_{m+1}\tau_1,& \tau_3\cdots\tau_{m+1},& \tau_3\cdots\tau_{m},& \dots,& \tau_3
		\end{array}\right)}.
	\end{align*}
	By substituting $y_1:=0$ and $y_i:=x_1+\cdots+x_{m+2-i}$ ($2\leqslant i\leqslant m+1$) and $\tau_1:=\sigma_1^{-1}$ and $\tau_2:=\sigma_m$ and $\tau_j:=\sigma_{m+2-j}\sigma_{m+3-j}^{-1}$ ($3\leqslant j\leqslant m+1$) for any $\sigma_1,\dots,\sigma_m\in\Gamma$, we have
	\begin{align*}
		M^m\varia{x_1,\ x_2,\ \dots,\ x_m}{\sigma_1,\ \sigma_2,\ \dots,\ \sigma_m}
		& = M^m{\scriptsize\left(\begin{array}{rrrr}
			-x_1-\cdots-x_m,& x_1,& \dots,& x_{m-1} \\
			\sigma_m^{-1},& \sigma_1\sigma_m^{-1},& \dots,& \sigma_{m-1}\sigma_m^{-1}
		\end{array}\right)} \\
		& = \mathrm{push}(M)^m\varia{x_1,\ x_2,\ \dots,\ x_m}{\sigma_1,\ \sigma_2,\ \dots,\ \sigma_m}.
	\end{align*}
	Whence we obtain the claim.
\end{proof}

The following generalizes
\cite[Theorem 14]{RS}.

\begin{prop}\label{prop:DKV2}
	For any mould $ M\in\mathrm{Fil}_{\mathcal{D}}^{2}\mathrm{ARI}(\Gamma)_{\underline{\mathrm{al}}/ \underline{\mathrm{al}}}$, its swap $\mathrm{swap}( M)$
	is pus-neutral \eqref{pus-neutral}, that is, for all $m\geqslant 1$ and $\sigma_1,\dots,\sigma_m\in\Gamma$,
	\begin{equation*}
		\sum_{i\in\mathbb{Z}/m\mathbb{Z}}\mathrm{pus}^i\circ\mathrm{swap}( M)^m
		\varia{\sigma_1,\dots,\sigma_m}{x_1,\dots,x_m}
		=0.
	\end{equation*}
\end{prop}

\begin{proof}

	For $m=0,1$, it is obvious because we have $M^1(x_1)=0$.

	Assume $m\geqslant2$. Let $\binom{\tau_0}{u_0},\binom{\tau_1}{u_1},\dots,\binom{\tau_m}{u_m}\in Y_{\mathbb{Z}}$ with $\tau_0\tau_1\cdots\tau_m=1$. By using alternality of $\mathrm{swap}(M)$, we have
	\begin{equation*}
		\sum_{\alpha\in Y_{\mathbb{Z}}^\bullet}
		\Sh{\scriptsize\left(\begin{array}{ccc}
			\tau_2,& \dots,& \tau_m \\
			u_2-u_1,& \dots,& u_m-u_1
		\end{array}\right)}{\scriptsize\left(\begin{array}{c}
			\tau_0 \\
			u_0-u_1
		\end{array}\right)}{\alpha}\mathrm{swap}(M)^m(\alpha)=0.
	\end{equation*}
	On the other hand, we calculate
	\begin{align*}
		&\sum_{\alpha\in Y_{\mathbb{Z}}^\bullet}
		\Sh{\scriptsize\left(\begin{array}{ccc}
			\tau_2,& \dots,& \tau_m \\
			u_2-u_1,& \dots,& u_m-u_1
		\end{array}\right)}{\scriptsize\left(\begin{array}{c}
		    \tau_0\\
			u_0-u_1
		\end{array}\right)}{\alpha}
		\mathrm{swap}(M)^m(\alpha) \\
		=&\undertilde{u}(M)^{m+1}\varia{\tau_1,\ \tau_2,\ \dots,\ \tau_m,\ \tau_0}
			{u_1,\ u_2,\ \dots,\ u_m,\ u_0}
		+\undertilde{u}(M)^{m+1}\varia{\tau_1,\ \tau_2,\ \dots,\ \tau_{m-1},\ \tau_0,\ \tau_m}
			{u_1,\ u_2,\ \dots,\ u_{m-1},\ u_0,\ u_m} \\
		&\hspace{0cm}+\cdots+\undertilde{u}(M)^{m+1}\varia{\tau_1,\ \tau_2,\ \tau_0,\ \tau_3,\ \dots,\ \tau_m}
			{u_1,\ u_2,\ u_0,\ u_3,\ \dots,\ u_m}
		+\undertilde{u}(M)^{m+1}\varia{\tau_1,\ \tau_0,\ \tau_2,\ \dots,\ \tau_m}
			{u_1,\ u_0,\ u_2,\ \dots,\ u_m}.
		\intertext{Repeated applications of \eqref{eqn:cyc} yield}
		=&\undertilde{u}(M)^{m+1}\varia{\tau_0,\ \tau_1,\ \tau_2,\ \dots,\ \tau_m}
			{u_0,\ u_1,\ u_2,\ \dots,\ u_m}
		+\undertilde{u}(M)^{m+1}\varia{\tau_0,\ \tau_m,\ \tau_1,\ \tau_2,\ \dots,\ \tau_{m-1}}
			{u_0,\ u_m,\ u_1,\ u_2,\ \dots,\ u_{m-1}} \\
		&\hspace{0cm}+\cdots+\undertilde{u}(M)^{m+1}\varia{\tau_0,\ \tau_3,\ \dots,\ \tau_m,\ \tau_1,\ \tau_2}
			{u_0,\ u_3,\ \dots,\ u_m,\ u_1,\ u_2}
		+\undertilde{u}(M)^{m+1}\varia{\tau_0,\ \tau_2,\ \dots,\ \tau_m,\ \tau_1}
			{u_0,\ u_2,\ \dots,\ u_m,\ u_1} \\
		=&\sum_{i\in\mathbb{Z}/m\mathbb{Z}}\undertilde{u}(M)^{m+1}\varia{\tau_0,\ \tau_{i+1},\ \tau_{i+2},\ \dots,\ \tau_{i+m}}
			{u_0,\ u_{i+1},\ u_{i+2},\ \dots,\ u_{i+m}}.
	\end{align*}
	Therefore, by substituting $u_0=0$ and $u_i=x_i$ ($1\leqslant i\leqslant m$) and $\tau_0=(\sigma_1\cdots\sigma_m)^{-1}$ and $\tau_i=\sigma_i$ ($1\leqslant i\leqslant m$), we get the claim.
\end{proof}

\begin{thm}\label{thm:embedding}
There is an embedding
$$
\mathrm{Fil}_{\mathcal{D}}^{2}\mathrm{ARI}(\Gamma)_{\underline{\mathrm{al}}/ \underline{\mathrm{al}}}\hookrightarrow
\mathrm{ARI}(\Gamma)_{\mathrm{push}/ \mathrm{pusnu}}
$$
of  graded Lie algebras.
\end{thm}

\begin{proof}
	Let $ M\in\mathrm{Fil}_{\mathcal{D}}^{2}\mathrm{ARI}(\Gamma)_{\underline{\mathrm{al}}/ \underline{\mathrm{al}}}$. Then we have $M\in\mathrm{ARI}(\Gamma)_{\underline\mathrm{al}/\underline\mathrm{al}}$. So by Proposition \ref{prop:DKV1} and Proposition \ref{prop:DKV2},  $M$ is $\mathrm{push}$-invariant
	and $\mathrm{swap}(M)$ is $\mathrm{pus}$-neutral.
	Therefore, we obtain $ M\in\mathrm{ARI}(\Gamma)_{\mathrm{push}/ \mathrm{pusnu}}$.
	To see that it is a Lie algebra homomorphism is immediate.
\end{proof}

As a corollary,
by taking an intersection with $\mathrm{ARI}(\Gamma)_\mathrm{al}^{\mathrm{fin},\mathrm{pol}}$ in the embedding of the above theorem,
we obtain the following  inclusion
which generalizes \cite[Theorem 3]{RS}.

\begin{cor}\label{cor:embedding}
There is an embedding
$$
\mathrm{Fil}_{\mathcal{D}}^{2}{\mathbb D}(\Gamma)_{\bullet\bullet}\hookrightarrow\mathfrak{lkrv}(\Gamma)_{\bullet\bullet}
$$
of bigraded Lie algebras.
\end{cor}

\begin{proof}
It follows from  Theorem \ref{thm:reform:krv:bigrade},
\eqref{eq:Fil2D=Fil2ARIalal}
and Theorem \ref{thm:embedding}.
\end{proof}

By imposing the distribution relation, we also obtain the following.

\begin{cor}\label{cor:embedding2}
There is an embedding
$$
\mathrm{Fil}_{\mathcal{D}}^{2}{D}(\Gamma)_{\bullet\bullet}\hookrightarrow\mathfrak{lkrvd}(\Gamma)_{\bullet\bullet}
$$
of bigraded Lie algebras.
\end{cor}

\begin{proof}
It follows from
Corollary \ref{cor:reform:lkrvd},
Corollary \ref{cor:reform:dihedral}
and Theorem \ref{thm:embedding}.
\end{proof}

It looks interesting to see if the map is an isomorphism.

\begin{rem}
By \eqref{eq:Fil2D=Fil2ARIalal}, Theorem \ref{thm:reform:krv:bigrade},
Theorem \ref{thm:embedding} and Corollary \ref{cor:embedding},
we obtain the commutative diagram \eqref{CD:D} of Lie algebras:
\begin{equation}\label{CD:D}
\xymatrix{
\mathrm{Fil}_{\mathcal{D}}^{2}{\mathbb D}(\Gamma)_{\bullet\bullet}\ar^{\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\simeq}[r]\ar@{^{(}->}[d]& \mathrm{Fil}_{\mathcal{D}}^{2}\mathrm{ARI}(\Gamma)_{\underline{\mathrm{al}}/ \underline{\mathrm{al}}}^{\mathrm{fin},\mathrm{pol}}\ar@{^{(}->}[d]\\
\mathfrak{lkrv}(\Gamma)_{\bullet\bullet}
\ar^{\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\simeq}[r]&\mathrm{ARI}(\Gamma)_{\mathrm{push}/ \mathrm{pusnu}}\cap \mathrm{ARI}(\Gamma)_\mathrm{al}^{\mathrm{fin},\mathrm{pol}}
}
\end{equation}
\end{rem}

redpajama_set_name: RedPajamaArXiv