text:
\section{Introduction}

Let $V$ be a finite-dimensional representation of the orthogonal Lie algebra $D_n$.  Then, we may define an abelian extension of $D_n$ by $D_n \inplus V$:  The representation action of $D_n$ on $V$  defines a multiplication between  elements of $D_n$ and $V$, and $V$ is regarded as an abelian algebra:  $[V,V]=0$.

 In this article, we determine all abelian extensions of $D_n$ that may be embedded into the exceptional Lie algebra $E_{n+1}$, $n=5, 6$, and $7$ (Section \ref{sec6}).  We then classify these embeddings, up to inner automorphism (Section \ref{sec7}).   As an application, we also consider the restrictions to $D_n \inplus V$ of irreducible representations of $E_{n+1}$, for each abelian extension and each embedding (Section \ref{sec8}).

We begin in Section \ref{sec2} with motivation.  Section \ref{sec3} records relevant information about the objects of study in this article:  $D_n$ and $E_{n+1}$, $n=5, 6$, $7$.  Section \ref{sec4} presents pertinent notation and terminology.  In Section \ref{sec5} we describe the classification of  embeddings of $D_n$ into $E_{n+1}$, up to inner automorphism, which will be employed in the following sections.

\section{Motivation}\label{sec2}

The examination of embeddings of semisimple Lie algebras into complex semisimple Lie algebras is an interesting, important, and widely studied problem.  The foundational work of Dynkin \cite{dynkin} is perhaps the most widely cited.  Dynkin's work \cite{dynkin}, together with a recent article by Minchenko \cite{min}, classifies the semisimple subalgebras
of the exceptional Lie algebras, up to inner automorphism.  From a classification of subalgebras, one may obtain a classification of embeddings in a straightforward manner.  In particular, a subalgebra $\mathfrak{g}'$ of $\mathfrak{g}$ naturally corresponds to one embedding, and may (or may not) correspond to more inequivalent embeddings if $\mathfrak{g}'$ has outer automorphisms.

The current paper extends previous research by the likes of Dynkin and Minchenko by considering embeddings of {\it nonsemisimple} Lie algebras into  exceptional Lie algebras.  It also extends and generalizes  previous work of Douglas and Repka, which examined specific important examples of embeddings of nonsemisimple Lie algebras into classical Lie algebras.

For instance, in \cite{dr3}, the authors classified the embeddings of
the Euclidean algebra $\mathfrak{e}(3)\cong \mathfrak{so}(3,\mathbb{C}) \inplus \mathbb{C}^3$ into $\mathfrak{so}(5,\mathbb{C})$, up to inner automorphism.  As another example, the authors  classified the embeddings of $\mathfrak{e}(3)$ into $\mathfrak{sl}(4,\mathbb{C})$, up to inner automorphism \cite{dr1}.

Little is known about the indecomposable representations of abelian extensions $D_n \inplus V$.  The current research promises to shed some light on the representation theory of these nonsemisimple Lie algebras.   In particular,   as an application of the above mentioned classification,  the authors will examine the irreducible representations of $E_{n+1}$ restricted to $D_n \inplus V$, for each embedding and each abelian extension.

The authors have employed this technique in previous research.  In \cite{dr3}, for instance,  Douglas and Repka showed that  each irreducible representation of $B_2$ remains indecomposable upon restriction to $\mathfrak{e}(3)$ $\cong$ $\mathfrak{so}(3,\mathbb{C}) \inplus \mathbb{C}^3$ with respect to any embedding of $\mathfrak{e}(3)$ into $B_2$,  hence creating a large family of indecomposable $\mathfrak{e}(3)$-representations.   In \cite{dr1} and \cite{dr2}   Douglas and Repka examined the irreducible representations of $A_3$ restricted to $\mathfrak{e}(3)$ with respect to any embedding;  these $\mathfrak{e}(3)$-representations were generally not indecomposable.

\section{The Lie algebras $D_n$ and $E_{n+1}$}\label{sec3}

The special orthogonal algebra $D_n$ is the Lie algebra of complex $2n\times 2n$ matrices $N$ satisfying $N^{tr}=-N$.  The dimension of $D_n$ is $2n^2-n$ and its rank is $n$.
The Lie group corresponding to  $D_n$ arises naturally as  the symmetry group of  a real projective space over $\mathbb{R}$.

Besides the classical Lie algebras, which include $D_n$, there are five exceptional Lie algebras, three of which are $E_6$, $E_7$, and $E_8$.  These exceptional Lie algebras  were first explicitly constructed by Cartan in his $1894$ thesis, although their existence was proposed several years earlier by Wilhelm Killing \cite{baez}.

Like $D_n$, the exceptional Lie algebras  $E_6$, $E_7$, and $E_8$  have a close connection to  the Riemannian geometry of projective spaces (for details, we refer the reader to \cite{baez}).  The algebras $E_6$, $E_7$, and $E_8$ have ranks $6$, $7$, and $8$, and dimensions $78$, $133$, and $248$, respectively.

Let $\mathfrak{g}$ denote $D_n$, $E_6$, $E_7$, or $E_8$.
Let $k=n, 6, 7$, or $8$ for $D_n$, $E_6$, $E_7$, or $E_8$, respectively.  We may define $\mathfrak{g}$ by a
set of generators $\{H_i, X_i, Y_i\}_{1\leq i \leq k}$ together with the Chevalley-Serre relations \cite{humphreys}:
\begin{equation}\label{cbd}
\begin{array}{llllllllllll}
\displaystyle [H_i,H_j]=0, &[H_i,X_j]= M^{\mathfrak{g}}_{ji} X_j,  \\
\displaystyle [H_i,Y_j]= -M^{\mathfrak{g}}_{ji} Y_j,& [X_i,Y_j]= \delta_{ij} H_i, \\
\displaystyle  (\text{ad} ~X_i)^{1-M^{\mathfrak{g}}_{ji}}(X_{j}) =0,  & (\text{ad} ~Y_i)^{1-M^{\mathfrak{g}}_{ji}}(Y_j)=0,& \text{when}~i\neq j,
\end{array}
\end{equation}
where $1 \leq i,j \leq k$, and  $M^{\mathfrak{g}}$ is the Cartan matrix of $\mathfrak{g}$  (see \cite{humphreys}).   The $X_i$, for $1\leq i \leq k$, correspond to the simple roots.

There are numerous calculations throughout this article involving products of elements of $E_{n+1}$, $n=5, 6, 7$.  These calculations may be carried out with the Chevalley-Serre relations \eqref{cbd}, or,  more efficiently, using the computer algebra system GAP \cite{gap}.

The Dynkin diagrams of $D_n$, $E_6$, $E_7$, and $E_8$, indicating the numbering of simple roots, are given in Figure \ref{ddd}.  For future reference, we also list in Table \ref{exb} the maximal dimension of an abelian
subalgebra of $E_6$, $E_7$, and $E_8$ from \cite{mal}.

\begin{table} [!ht]\renewcommand{\arraystretch}{1.2} \caption{Maximal abelian subalgebras of $E_6$, $E_7$, and $E_8$. \cite{mal}} \label{exb}\begin{center}
\begin{tabular}{|c|c|c|clclcl}
\hline
Exceptional Lie Algebra &  Maximal Dimension of Abelian Subalgebra \\
\hline \hline
$E_6$& $16$\\
\hline
$E_7$  & $27$\\
\hline
$E_8$ & $36$\\
\hline
\end{tabular}\end{center}
\end{table}

\begin{figure} [!ht]\caption{Dynkin diagrams of $D_n$, $E_{n+1}$.}\label{ddd}
 \begin{align*}\begin{array}{ccc}
 D_n  &&\underset{\mathclap{1}}{\circ} - \underset{\mathclap{2}}{\circ} - \dotsb - \underset{\mathclap{{n-2}}}{\overset{\overset{\textstyle\circ_{\mathrlap{n-1}}}{\textstyle\vert}}{\circ}} \,-\, \underset{\mathclap{{n}}}{\circ} \\\\
E_{n+1}  &&\underset{\mathclap{1}}{\circ} - \underset{\mathclap{3}}{\circ}                 - \underset{\mathclap{4}}{\overset{\overset{\textstyle\circ_{\mathrlap{2}}}{\textstyle\vert}}{\circ}}                - \underset{\mathclap{5}}{\circ} - \dotsb - \underset{\mathclap{n+1}}{\circ}  \\\\
\end{array}
\end{align*}
\end{figure}

We now consider the representations of $
\mathfrak{g} =
D_n$
or
$E_{n+1}$, with
$k$ as above.

For $i=1,...,k$, and $\mathfrak{h}$ the Cartan subalgebra of $\mathfrak{g}$ with basis $H_1,...,H_k$, define
$\alpha_i, ~ \lambda_i \in \mathfrak{h}^*$ by $\alpha_i(H_j)= M^{\mathfrak{g}}_{ji}$, and $\lambda_i(H_j)=\delta_{ij}$.  The $\lambda_i$ are the {\it fundamental weights}, and their indexing corresponds with that  of the  Dynkin diagram of type $D_n$, $E_6$, $E_7$, or $E_8$, respectively, in  Figure \ref{ddd}.

For each
$\lambda= m_1 \lambda_1+...+m_k \lambda_k  \in \mathfrak{h}^*$, with nonnegative
integers $m_1,...,m_k$,
there exists a finite-dimensional,
irreducible $\mathfrak{g}$-module
with highest weight $\lambda$,
denoted $V_{\mathfrak{g}}(\lambda)$, and every finite-dimensional irreducible $\mathfrak{g}$-module is of this form, for some $\lambda$.  The representations $V_{\mathfrak{g}}(\lambda_i)$ for $1\leq i \leq k$ are the {\it fundamental
representations}.  A useful formula for the dimensions of the fundamental representations of $D_n$ is as follows [\cite{hu}, Theorem $13.10$]:
\begin{equation}
\dim (V_{D_n}(\lambda_m) )=  \begin{cases} \binom{2n}{m}, & \text{if} ~~ 1 \leq m< n-1;\\
2^{n-1}, & \text{if} ~~ m= n-1,n.
\end{cases}
\end{equation}
The representation $V_{\mathfrak{g}}(\lambda)$ is realized as the quotient of $\mathcal{U}(\mathfrak{g})$ by the left ideal, $J_{\lambda}$,
generated by $X_i, H_i- \lambda(H_i)$, $Y_{i}^{1+\lambda(H_i)}$, $1\leq i \leq k$ (here the action of $\mathcal{U}(\mathfrak{g})$ on itself and on
$V_{\mathfrak{g}}(\lambda)$ is given by left multiplication).  We will denote the element $1+J_\lambda$ of  the quotient $V_{\mathfrak{g}}(\lambda)$ by
$\widetilde{u}$.  Then one can show that $V_{\mathfrak{g}}(\lambda)$ is generated by $\{ Y_{i_1} \cdots Y_{i_l} \widetilde{u}: l \in \mathbb{N}_0, i_1,...,
i_l \in \{1,...,k\} \}$.  The weight of   $Y_{i_1} \cdots Y_{i_l} \widetilde{u}$ is $\lambda- \Sigma_{j=1}^l \alpha_{i_j}$.

\section{Additional definitions and notation}\label{sec4}

The following definitions and notation will be used in this article.  In the definitions, let $\mathfrak{g}$ and $\mathfrak{g}'$ be simple Lie algebras.

\begin{itemize}
\item Let $X_{a_i}$ correspond to a simple root of $E_{n+1}$, for $ 1\leq a_i \leq n+1$, where $n=5, 6$, or $7$.  We then define
\[
X_{a_1,a_2,a_3,...,a_k} \equiv [[...[[X_{a_{1}},X_{a_{2}}],X_{a_{3}}],...],X_{a_{k}}].
\]
$Y_{a_1,a_2,a_3,...,a_k}$ is defined analogously.

\item   Let $\varphi:  D_{n} \hookrightarrow E_{n+1}$ be an embedding, and let $W \in E_{n+1}$.     Then, we define $[W]_{\varphi(D_n)}$ to be the $D_n$-representation generated  by $W$ with respect to the adjoint action of $\varphi(D_n)$.

\item Let  $W_1$,..., $W_m$ $\in E_{n+1}$, for $n=5, 6$, or $7$. Then,
$\langle W_1,..., W_m \rangle_{E_{n+1}}$
is the subalgebra of $E_{n+1}$ generated by $W_1$,...,$W_m$.

\item Let $V$ be a finite-dimensional representation of $D_n$.
Then, a {\it lift} of the embedding $\varphi: D_n \hookrightarrow E_{n+1}$ to $D_n \inplus V$ is an embedding $\widetilde{\varphi}^V: D_n \inplus V\hookrightarrow E_{n+1}$  such that $\widetilde{\varphi}^V$ restricted to $D_n$ is equal to $\varphi$.  That is,   $\widetilde{\varphi}^V |_{D_n}= \varphi$.

\item Let $\varphi$ and $\varrho$ be Lie algebra embeddings of $\mathfrak{g}'$ into $\mathfrak{g}$.  Then $\varphi$ and $\varrho$ are {\it equivalent} if there is an inner automorphism $\rho: \mathfrak{g} \rightarrow \mathfrak{g}$ such that $\varphi= \rho \circ
\varrho$, and we write
\[
\varphi \sim \varrho.
\]
Hence, our classification in this article is up to equivalence.
\item Two embeddings $\varphi$ and $\varrho$  of $\mathfrak{g}'$ into $\mathfrak{g}_{}$ are \emph{linearly equivalent} if for any finite-dimensional  representation $V$ of $\mathfrak{g}$,  $V|_{\varphi(\mathfrak{\mathfrak{g}'})}$ and $V|_{\varrho(\mathfrak{\mathfrak{g}'})}$ are isomorphic representations of $\mathfrak{g}'$, and we write
\[
\varphi \sim_L \varrho.
\]
Clearly equivalence implies linear equivalence, but the converse is not in general true.   We define equivalence  of subalgebras much as we did for embeddings.
\item Two subalgebras $\mathfrak{g}'$ and $\mathfrak{g}''$ of $\mathfrak{g}$ are equivalent if there is an inner automorphism $\rho$ or $\mathfrak{g}$
such that $\rho(\mathfrak{g}')=\mathfrak{g}''$.

\item A subalgebra of $\mathfrak{g}$ is {\it regular} if it is normalized by a Cartan subalgebra of $\mathfrak{g}$.
\end{itemize}

 \section{Embeddings of $D_{n}$ into $E_{n+1}$}\label{sec5}

In this section we describe the classification of  embeddings of $D_n$ into $E_{n+1}$, up to equivalence, which is given
in Theorem \ref{dymin}.  We begin by constructing two embeddings of $D_n$ into $E_{n+1}$.

With respect to the generators defined in Section \ref{sec3}, we may ``naturally" embed $D_{n}$ into $E_{n+1}$ for $n=5, 6,$ or $7$ as follows:
\begin{equation}
\begin{array}{llll}\label{emb}
\varphi_n: &D_n &\hookrightarrow& E_{n+1} \\
& H_{n+1-i} & \mapsto & H_{i+1}\\
& X_{n+1-i} & \mapsto & X_{i+1}\\
& Y_{n+1-i} & \mapsto & Y_{i+1},
\end{array}
\end{equation}
where $1\leq i\leq n$.

Each of these embeddings may be  visualized as arising from a ``natural" subgraph of the Dynkin diagram of $E_{n+1}$, for $n=5,6,$ or $7$, which is isomorphic to the Dynkin diagram of $D_n$  (see Figure \ref{ddd}).

Consider the transposition $\sigma =(n-1,n)$  of the symmetric group $\mathcal{S}_n$, and define an outer automorphism of $D_n$ as follows:
\begin{equation}
\begin{array}{llll}\label{emb2}
\rho_n: &D_n &\rightarrow& D_{n} \\
& H_{i} & \mapsto & H_{\sigma(i)}\\
& X_{i} & \mapsto & X_{\sigma(i)}\\
& Y_{i} & \mapsto & Y_{\sigma(i)},\\
\end{array}
\end{equation}
where $1\leq i\leq n$.    The outer automorphisms of $D_{n}$ correspond to the automorphisms of its Dynkin diagram [\cite{fh}, Proposition $D. 40$], so that $\rho_n$ is the only outer automorphism of $D_{n}$, $n>4$,  up to inner automorphism.

Using the above outer automorphism of $D_n$, define a second embedding of $D_n$ into $E_{n+1}$:
\begin{equation}
\varrho_n \equiv \varphi_n \circ \rho_n :  D_n \hookrightarrow E_{n+1}.
\end{equation}

The two embeddings $\varphi_n$ and $\varrho_n$ are the basis of the classification of embeddings of $D_n$ into $E_{n+1}$.  The classification is  a consequence of the following three theorems.

\begin{theorem}\cite{dynkin, min} \label{dymina}
There is a unique $D_n$ subalgebra inside $E_{n+1}$, up to equivalence, for $n=5,6$, and $7$.  Further, in each case, the subalgebra is regular.
\end{theorem}

\begin{theorem}[\cite{min}, Theorem $2$] \label{dymin2}
Let $\varphi$ and $\varrho$ be embeddings of $D_n$ into $E_{n+1}$, for $n=5,6$, and $7$, then
 \begin{equation}\label{vccd}
 \varphi \sim_L \varrho \Leftrightarrow V_{E_{n+1}}(\lambda_{n+1})|_{\varphi(D_n)}\cong V_{E_{n+1}}(\lambda_{n+1})|_{\varrho(D_n)}.
 \end{equation}
\end{theorem}

Interestingly, Theorem \ref{dymin2}, from \cite{min}, is stronger than a corresponding result found in Dynkin's classic paper [\cite{dynkin}, Theorem $1.3$].  In \cite{dynkin}, one needs to consider $V_{E_{n+1}}(\lambda_{n+1})$, and $V_{E_{n+1}}(\lambda_{1})$, for $n=6, 7$, to establish linear equivalence.  Note that Theorem \ref{dymin2}, although found in \cite{min}, is attributed to Losev \cite{losev}.
\begin{theorem}[\cite{min}, Theorem $3$] \label{dymin2b}
Let  $\varphi$ and $\varrho$ be embeddings of the simple Lie algebra $\mathfrak{g}'$ into
the
simple Lie algebra $\mathfrak{g}$ such that $\varphi(\mathfrak{\mathfrak{g}'})$ and $\varrho(\mathfrak{\mathfrak{g}'})$ are regular subalgebras.  Then
 \begin{equation}\label{vcc}
 \varphi \sim_L \varrho \Leftrightarrow  \varphi \sim \varrho.
 \end{equation}
\end{theorem}

\begin{theorem}\label{dymin}
The  embedding $\varphi_7$ is the unique embedding of $D_7$ into $E_{8}$, up to equivalence.
 There are exactly two embeddings of  $D_5$ into $E_{6}$ and of $D_6$ into $E_{7}$, up to equivalence: $\varphi_n$ and $\varrho_n$, for $n=5$ or $6$, respectively.
\end{theorem}
\begin{proof}
We first show that $\varphi_n: D_n \hookrightarrow E_{n+1}$ and $\varrho_n: D_n \hookrightarrow E_{n+1}$ are equivalent if and only if $n=7$.  Consider the following decompositions.
   \begin{equation}\label{wq1}
\begin{array}{llllllll}
 V_{E_6}(\lambda_6)|_{\varphi(D_5)}
&\cong&   V_{D_5}(0) \oplus  V_{D_5}(\lambda_1)\oplus V_{D_5}(\lambda_5),\\
V_{E_6}(\lambda_6)|_{\varrho (D_5)}
&\cong  & V_{D_5}(0) \oplus  V_{D_5}(\lambda_1)\oplus V_{D_5}(\lambda_4).
\end{array}
 \end{equation}
$V_{E_6}(\lambda_6)|_{\varphi(D_5)} \ncong  V_{E_6}(\lambda_6)|_{\varrho (D_5)}$ implies $\varphi_5 \nsim_L \varrho_5$, and hence
    \begin{equation}\label{w1}
 \varphi_5 \nsim \varrho_5.
 \end{equation}
 Also,
  \begin{equation}\label{wq2}
\begin{array}{llllllll}
 V_{E_7}(\lambda_7)|_{\varphi(D_6)}
&\cong&   2V_{D_6}(\lambda_1)\oplus V_{D_6}(\lambda_6),\\
V_{E_7}(\lambda_7)|_{\varrho (D_6)}
&\cong  &  2V_{D_6}(\lambda_1)\oplus V_{D_6}(\lambda_5).
\end{array}
 \end{equation}
$V_{E_7}(\lambda_7)|_{\varphi(D_6)} \ncong  V_{E_7}(\lambda_7)|_{\varrho (D_6)}$  implies $\varphi_6 \nsim_L \varrho_6$, and hence
  \begin{equation}\label{w2}
 \varphi_6 \nsim \varrho_6.
 \end{equation}
 Finally,
 \begin{equation}\label{wq3}
\begin{array}{llllllllllllll}
 V_{E_8}(\lambda_8)|_{\varphi(D_7)} \cong  V_{E_8}(\lambda_8)|_{\varrho (D_7)}&\cong    V_{D_7}(\lambda_2) \oplus V_{D_7}(\lambda_1)  \oplus
  V_{D_7}(\lambda_6)\\
 &\oplus V_{D_7}(\lambda_7) \oplus
 V_{D_7}(\lambda_1)\oplus V_{D_7}(0).
\end{array}
 \end{equation}
 $V_{E_8}(\lambda_8)|_{\varphi(D_7)} \cong  V_{E_8}(\lambda_8)|_{\varrho (D_7)}$
 implies $\varphi_7 \sim_L \varrho_7$ (Theorem \ref{dymin2}), and hence, by Theorem  \ref{dymin2b},
 \begin{equation}\label{w3}
 \varphi_7 \sim \varrho_7.
 \end{equation}
 Since, by Theorem \ref{dymina}, there is only one $D_n$ subalgebra in $E_{n+1}$, up to equivalence, the only possible embedding inequivalent to $\varphi_n$ is formed from composing $\varphi_n$ with an outer automorphism of $D_n$, of which there is only one for $n>4$, namely $\rho_n$ \cite{fh}.  In other words, the only possible embedding inequivalent to $\varphi_n$ is $\varrho_n$.  The result then follows from Eqs. \eqref{w1}, \eqref{w2}, and \eqref{w3}.
 \end{proof}

\section{Abelian extensions of ${D}_n$ in $E_{n+1}$}\label{sec6}

In this section we determine the abelian extensions of $D_n$ that may be embedded into $E_{n+1}$, $n=5, 6$, $7$.   The results are summarized in Propositions \ref{mhh}, \ref{mhhh}, and \ref{mhhhh}.

\subsection{Abelian extensions of  ${D}_7$ in $E_{8}$}
We begin by calculating the decomposition of $E_8$ with respect to the adjoint action of $\varphi_7(D_7)$:
\begin{equation}\label{hhhhj}
\begin{array}{lllllllllllll}
E_8&\cong_{\varphi_7(D_7)}& [  X']_{\varphi_7(D_7)}&\oplus&  [ X''' ]_{\varphi_7(D_7)} &\oplus&  [ Y_{1} ]_{\varphi_7(D_7)} &\oplus\\
&&  [ X'']_{\varphi_7(D_7)} &\oplus&  [ Y' ]_{\varphi_7(D_7)}&\oplus&
[ H ]_{\varphi_7(D_7)}\\
&  \cong_{\varphi_7(D_7)} & V_{D_7}(\lambda_2) &\oplus& V_{D_7}(\lambda_1) &\oplus& V_{D_7}(\lambda_6)&\oplus  \\
&& V_{D_7}(\lambda_7) &\oplus& V_{D_7}(\lambda_1)&\oplus&  V_{D_7}(0),
\end{array}
\end{equation}
where
\begin{equation}\label{ghyz}
\begin{array}{llll}
\displaystyle  X'&=&X_{4,5,6,7,8,2,3,4,5,6,7},\\
\displaystyle  X'' &=&-X_{3,4,2,1,5,4,3,6,5,4,7,2,6,5,8,7,6,4,5,3,4,2}, \\
\displaystyle  X'''&=&  X_{8,7,6,5,4,3,2,1,4,5,6,7,3,4,5,6,2,4,5,3,4,2,1,3, 4,5,6,7,8},\\% bE8[1][120]
\displaystyle Y'&=& -Y_{5,4,2,3,6,4,1,3,5,4,7,2,6,5,4,3,1},\\
\displaystyle H&=&  4H_1+5H_2+7H_3+10H_4+\\
\displaystyle && 8H_5+6H_6+4H_7+2H_8.
\end{array}
\end{equation}
Note that
\begin{equation} \label{idd}
\varphi_7(D_7) = [X']_{\varphi_7(D_7)} \cong V_{D_7}(\lambda_2).
\end{equation}
Since $\varphi_7$ is the only embedding of $D_7$ into $E_8$, up to equivalence (Theorem \ref{dymin}), determining which abelian extensions of $D_7$ may be embedded into $E_8$
amounts
to determining which $E_8$-subspaces among $[ X''' ]_{\varphi_7(D_7)}$, $[ Y_{1} ]_{\varphi_7(D_7)}$, $[ X'']_{\varphi_7(D_7)}$, $[ Y' ]_{\varphi_7(D_7)}$, and
$[ H ]_{\varphi_7(D_7)}$ in Eq. \eqref{hhhhj}, or direct sums of these subspaces, are abelian subalgebras of $E_8$.
\begin{lemma}\label{abal}
The $E_8$-subspaces  $[ X''']_{\varphi_7(D_7)}$,   $[ Y']_{\varphi_7(D_7)}$, and   $[ H ]_{\varphi_7(D_7)}$   are abelian subalgebras of $E_8$.
\end{lemma}
\begin{proof}
Let $X'''_{a_{1},a_{2},...,a_{m}}=  [Y_{a_n},[Y_{a_{n-1}},...,[Y_{a_2},[Y_{a_1},X''']]...]]$;  then the following is a basis of  $V_{D_7}(\lambda_1)\cong [ X''']_{\varphi_7(D_7)}$:
 \begin{equation}\label{asgg}
\begin{array}{lllllll}
X''', &X'''_{8},& X'''_{8,7}, \\
X'''_{8,7,6}, & X'''_{8,7,6,5}, & X'''_{8,7,6,5,4},\\
 X'''_{8,7,6,5,4,2}, & X'''_{8,7,6,5,4,3}, & X'''_{8,7,6,5,4,2,3},\\
 X'''_{8,7,6,5,4,2,3,4}, & X'''_{8,7,6,5,4,2,3,4,5}, & X'''_{8,7,6,5,4,2,3,4,5,6},\\
  X'''_{8,7,6,5,4,2,3,4,5,6,7}, & X'''_{8,7,6,5,4,2,3,4,5,6,7,8}.
\end{array}
\end{equation}
Direct calculation shows
\begin{equation}\label{pjii}
[[ X''' ]_{\varphi_7(D_7)}, X''']=0.
\end{equation}
Eq. \eqref{pjii} together with the Jacobi identity imply
\begin{equation}
[[ X''' ]_{\varphi_7(D_7)}, [ X''']_{\varphi_7(D_7)}]=0.
\end{equation}
Hence, $[ X''' ]_{\varphi_7(D_7)}$ is an abelian subalgebra of $E_8$.   In a similar fashion, we show $[ Y']_{\varphi_7(D_7)}$ is an abelian subalgebra of $E_8$.   $[ H ]_{\varphi_7(D_7)}$  is $1$-dimensional, hence is  necessarily abelian.
\end{proof}

\begin{lemma}\label{notab}
The subspaces $[\alpha X'''+ \beta Y']_{\varphi_7(D_7)}$, with $\alpha, \beta \in \mathbb{C}^*$,  $[X''']_{\varphi_7(D_7)}$ $\oplus$ $[Y']_{\varphi_7(D_7)}$,
$[X''']_{\varphi_7(D_7)} \oplus [H]_{\varphi_7(D_7)}$, $[H]_{\varphi_7(D_7)} \oplus [Y']_{\varphi_7(D_7)}$,  $[X''']_{\varphi_7(D_7)} \oplus [Y']_{\varphi_7(D_7)} \oplus [H]_{\varphi_7(D_7)}$,  $[Y_1]_{\varphi_7(D_7)}$, and $[X'']_{\varphi_7(D_7)}$ are not  abelian subalgebras of $E_8$.
\end{lemma}
\begin{proof}
Note that $Y_8 \in \varphi_7(D_7)$, so that $[Y_8,  \alpha X'''+ \beta Y'] \in [\alpha X'''+ \beta Y']_{\varphi_7(D_7)}$.  We have
\begin{equation}
[\alpha X'''+ \beta Y', [Y_8,  \alpha X'''+ \beta Y'] ] = -2\alpha \beta X' \neq 0.
\end{equation}
Thus, $[\alpha X'''+ \beta Y']_{\varphi_7(D_7)}$ is not abelian.  In a similar manner, we show that $[X''']_{\varphi_7(D_7)} \oplus [Y']_{\varphi_7(D_7)}$,
$[X''']_{\varphi_7(D_7)} \oplus [H]_{\varphi_7(D_7)}$, $[H]_{\varphi_7(D_7)} \oplus [Y']_{\varphi_7(D_7)}$, and $[X''']_{\varphi_7(D_7)} \oplus [Y']_{\varphi_7(D_7)} \oplus [H]_{\varphi_7(D_7)}$ are not abelian.

Finally,
$[Y_1]_{\varphi_7(D_7)}$ and $[X'']_{\varphi_7(D_7)}$ are not  abelian subalgebras of $E_8$ by Table \ref{exb} since
$\dim(V_{\varphi(D_7)}(\lambda_6))$ $=$
$\dim(V_{\varphi(D_7)}(\lambda_7))=64$.
\end{proof}

Note the following isomorphisms of $\varphi(D_7)$-modules:
\begin{equation}\label{labb}
\begin{array}{llllllll}
\displaystyle [H]_{\varphi_7(D_7)} &\cong& V_{D_7}(0) &\cong&  \mathbb{C},\\
\displaystyle [X''']_{\varphi_7(D_7)} &\cong& V_{D_7}(\lambda_1)&\cong& \mathbb{C}^{14},\\
\displaystyle [Y']_{\varphi_7(D_7)} &\cong& V_{D_7}(\lambda_1)&\cong& \mathbb{C}^{14}.
\end{array}
\end{equation}
The irrep $V_{D_{7}}(\lambda_{1})$ is the ``standard'' representation of $D_{7} \cong \mathfrak{so}(14,\mathbb{C})$ on
${\Bbb C}^{14}$ by matrix multiplication.  We shall write ${\Bbb C}^{14}$ to refer to this irrep,
and ${\Bbb C}$ to refer to the trivial irrep $V_{D_{7}}(0)$.

Lemmas \ref{abal} and \ref{notab}  then give the following proposition.
\begin{proposition}\label{mhh}
The only abelian extensions of $D_7$ that may be embedded into $E_8$ are the following:
\begin{equation}
\begin{array}{ll}
D_7 \inplus \mathbb{C} ~ \text{and} ~ D_7 \inplus \mathbb{C}^{14}.
\end{array}
\end{equation}
The abelian subspaces of $E_{8}$ which are isomorphic to ${\Bbb C}$ with respect to the adjoint action of
$\varphi_7(D_7)$ have (highest weight) vector $\alpha  H$, for $\alpha \in \mathbb{C}^*$.  Those
which are isomorphic to $\mathbb{C}^{14}$ with respect to the adjoint action of $\varphi_7(D_7)$
have highest weight vector
$\alpha  X'''$ or $\alpha  Y'$, for $\alpha \in \mathbb{C}^*$.
\end{proposition}

\subsection{Abelian extensions of $D_6$ in $E_7$}

We begin by calculating the decomposition of $E_7$ with respect to the adjoint action of $\varphi_6(D_6)$ and $\varrho_6(D_6)$, respectively:
\begin{equation}\label{hhhh}
\begin{array}{lllllllll}
E_7 &
\cong_{\varphi_6(D_6)}&  [ X']_{\varphi_6(D_6)} &\oplus& [ X'' ]_{\varphi_6(D_6)} &\oplus& [ Y_1 ]_{\varphi_6(D_6)} &\oplus& \\
&& [ Y' ]_{\varphi_6(D_6)} &\oplus& [ X''' ]_{\varphi_6(D_6)}& \oplus &
 [ H]_{\varphi_6(D_6)} \\
&\cong_{\varphi_6(D_6)} &V_{D_6}(\lambda_2) &\oplus&     2V_{D_6}(\lambda_5) &\oplus& 3 V_{D_6}(0),
\end{array}
\end{equation}
\begin{equation}\label{hhhhp}
\begin{array}{lllllllllll}
E_7 &
\cong_{\varrho_6(D_6)}&  [ X']_{\varrho_6(D_6)} &\oplus& [ X'' ]_{\varrho_6(D_6)} &\oplus& [ Y_1 ]_{\varrho_6(D_6)} &\oplus& \\
&& [ Y' ]_{\varrho_6(D_6)} &\oplus& [ X''' ]_{\varrho_6(D_6)}& \oplus&
 [ H]_{\varrho_6(D_6)} \\
&\cong_{\varrho_6(D_6)} &V_{D_6}(\lambda_2) &\oplus  &   2V_{D_6}(\lambda_6)& \oplus &3 V_{D_6}(0),
\end{array}
\end{equation}
where
\begin{equation}\label{ghyy}
\begin{array}{lllllll}
\displaystyle  X'&=& X_{6,7,5,4,3,2,4,5,6},\\
\displaystyle  X''&=& X_{7,6,5,4,3,2,4,5,6,1,3,4,5,2,4,3}, \\
\displaystyle  X'''&=& -X_{7,6,5,4,3,2,4,5,6,1,3,4,5,2,4,3,1},\\
\displaystyle  Y'&= &-Y_{7,6,5,4,3,2,4,5,6,1,3,4,5,2,4,3,1}, \\
\displaystyle  H&= &2H_1+2H_2+3H_3+4H_4+3H_5+2H_6+H_7.
\end{array}
\end{equation}
Note that
\begin{equation} \label{idddd}
\begin{array}{lllll}
\varphi_6(D_6) &=& [X']_{\varphi_6(D_6)} &\cong& V_{D_6}(\lambda_2),\\
\varrho_6(D_6) &=& [X']_{\varrho_6(D_6)} &\cong& V_{D_6}(\lambda_2).
\end{array}
\end{equation}
Since $\varphi_6$ and $\varrho_6$ are the only embeddings of $D_6$ into $E_7$, up to equivalence (Theorem \ref{dymin}),
determining which abelian extensions of $D_6$ may be embedded into $E_7$ amounts to determining which $E_7$-subspaces in Eq. \eqref{hhhh} or \eqref{hhhhp} (excluding $[X']_{\varphi_6(D_6)}$ and $[X']_{\varrho_6(D_6)}$), or direct sums of these subspaces, are abelian subalgebras of $E_7$.

Eqs. \eqref{hhhh} and \eqref{hhhhp} imply the following lemma.
\begin{lemma}\label{notabb}
With respect to the adjoint action of $\varphi_6(D_6)$ or $\varrho_6(D_6)$,
all
highest weight vectors in $E_7$
having weight zero are of the form  $\alpha Y' +\beta X''' +\gamma  H$, for $\alpha, \beta, \gamma \in \mathbb{C}$, not all zero.
\end{lemma}
\begin{lemma}\label{notabbc}
$[\alpha Y' +\beta X''' +\gamma  H]_{\varphi_6(D_6)}$  and  $[\alpha Y' +\beta X''' +\gamma  H]_{\varrho_6(D_6)}$ are abelian  subalgebras of $E_7$.   Further, any other subspace in the decomposition  of Eq. \eqref{hhhh} or \eqref{hhhhp}, or direct sums of such subspaces, is not an abelian subalgebra of $E_7$.
\end{lemma}
\begin{proof}
$[\alpha Y' +\beta X''' +\gamma  H]_{\varphi_6(D_6)}$  and  $[\alpha Y' +\beta X''' +\gamma  H]_{\varrho_6(D_6)}$ are $1$-dimensional, and hence necessarily abelian.

The subspaces $[X'']_{\varphi_6(D_6)}, [X'']_{\varrho_6(D_6)}$, $[Y_1]_{\varphi_6(D_6)}$, and  $[Y_1]_{\varrho_6(D_6)}$  are not abelian by Table \ref{exb} since $\dim(V_{D_6}(\lambda_5))$ $=$   $\dim(V_{D_6}(\lambda_6))$ $=$$32>27$.

$[X']_{\varphi_6(D_6)}$ and $[X']_{\varrho_6(D_6)}$ are isomorphic to $D_6$ and hence not abelian.  Noting $[Y', X'''] \neq 0, [Y',H] \neq 0$, and $[X''',H] \neq 0$  completes the proof.
\end{proof}
For $\alpha, \beta, \gamma \in \mathbb{C}$, not all zero, note the following isomorphisms:
\begin{equation}\label{labbb}
\begin{array}{llllllll}
\displaystyle [\alpha Y' +\beta X''' +\gamma  H]_{\varphi_6(D_6)} &\cong& \mathbb{C},\\
\displaystyle [\alpha Y' +\beta X''' +\gamma  H]_{\varrho_6(D_6)}  &\cong&  \mathbb{C}.
\end{array}
\end{equation}
Lemma \ref{notabb} and \ref{notabbc} then give us the following proposition.

\begin{proposition}\label{mhhh}
The only abelian extension of $D_6$ that may be embedded into $E_7$ is the following:
\begin{equation}
\begin{array}{ll}
D_6 \inplus \mathbb{C}.
\end{array}\end{equation}
The (abelian) subspaces of $E_{7}$ which are isomorphic to ${\Bbb C}$  with respect to the adjoint action
of $\varphi_6(D_6)$ or $\varrho_6(D_6)$ have (highest weight) vector
$\alpha Y' +\beta X''' +\gamma  H$, for $\alpha, \beta, \gamma \in \mathbb{C}$, not all zero.
\end{proposition}

\subsection{Abelian extensions of $D_5$ in $E_6$}

We begin by calculating the decomposition of $E_6$ with respect to the adjoint action of $\varphi_5(D_5)$ and $\varrho_5(D_5)$:
\begin{equation}\label{iiii}
\begin{array}{lllllllllll}
\displaystyle E_6 &\cong_{\varphi_5(D_5)}& [X']_{\varphi_5(D_5)}&\oplus& [Y_1]_{\varphi_5(D_5)}&\oplus& [X'']_{\varphi_5(D_5)} &\oplus&  \\
\displaystyle&& [H]_{\varphi_5(D_5)}\\
\displaystyle&\cong_{\varphi_5(D_5)}&  V_{D_5}(\lambda_2) &\oplus& V_{D_5}(\lambda_4) &\oplus& V_{D_5}(\lambda_5)&\oplus&  \\
\displaystyle &&V_{D_5}(0),
\end{array}
\end{equation}
\begin{equation}\label{iiiib}
\begin{array}{lllllllllll}
\displaystyle E_6 &\cong_{\varrho_5(D_5)}& [X']_{\varrho_5(D_5)}&\oplus& [Y_1]_{\varrho_5(D_5)}&\oplus& [X'']_{\varrho_5(D_5)} &\oplus & \\
\displaystyle &&[H]_{\varrho_5(D_5)}\\
\displaystyle &\cong_{\varrho_5(D_5)}&  V_{D_5}(\lambda_2) &\oplus& V_{D_5}(\lambda_5) &\oplus& V_{D_5}(\lambda_4)&\oplus & \\
\displaystyle &&V_{D_5}(0),
\end{array}
\end{equation}
where
\begin{equation}\label{ghyyy}
\begin{array}{llllll}
\displaystyle  X'&=&X_{6,5,4,3,2,4,5},\\
\displaystyle  X''&=&X_{6,5,4,2,3,1,4,3,5,4,2},\\
\displaystyle  H&=&2H_1+\frac{3}{2}H_2+\frac{5}{2}H_3+3H_4+2H_5+H_6.\\
\end{array}
\end{equation}
Note that
\begin{equation} \label{iddd}
\begin{array}{lllll}
\varphi_5(D_5) &=& [X']_{\varphi_5(D_5)} &\cong& V_{D_5}(\lambda_2),\\
\varrho_5(D_5) &=& [X']_{\varrho_5(D_5)} &\cong& V_{D_5}(\lambda_2).
\end{array}
\end{equation}

Since $\varphi_5$ and $\varrho_5$ are the only embeddings of $D_5$ into $E_6$, up to equivalence (Theorem \ref{dymin}),
determining which abelian extensions of $D_5$ may be embedded into $E_6$ amounts to determining which $E_6$-subspaces in Eq. \eqref{iiii} or \eqref{iiiib} (excluding $[X']_{\varphi_5(D_5)}$ and $[X']_{\varrho_5(D_5)}$), or direct sums of these subspaces, are abelian subalgebras of $E_6$.

The proofs of the following two lemmas proceed as in the above subsections and are omitted.
\begin{lemma}\label{aballl}
The $E_6$-subspaces  $[ Y_1]_{\varphi_5(D_5)}$,   $[ X'']_{\varphi_5(D_5)}$, and $[ H ]_{\varphi_5(D_5)}$, as well as $[ Y_1]_{\varrho_5(D_5)}$,   $[ X'']_{\varrho_5(D_5)}$, and $[ H ]_{\varrho_5(D_5)}$ are abelian subalgebras of $E_6$.
\end{lemma}

\begin{lemma}\label{notabbb}
The direct sum of any two or more of the subspaces in the decomposition  of Eq. \eqref{iiii} or \eqref{iiiib} is not an abelian subalgebra of $E_6$.
\end{lemma}

Note the following isomorphism of $D_5$-modules, with respect to the adjoint action of $\varphi_5(D_5)$ or $\varrho_5(D_5)$, respectively:
\begin{equation}\label{labbbb}
\begin{array}{ll}
\displaystyle [H]_{\varphi_5(D_5)} \cong   [H]_{\varrho_5(D_5)} \cong  \mathbb{C}.
\end{array}
\end{equation}
Lemmas \ref{aballl} and \ref{notabbb} then give the following proposition.
\begin{proposition}\label{mhhhh}
The only abelian extensions of $D_5$ that may be embedded into $E_6$ are the following:
\begin{equation}
\begin{array}{ll}
D_5 \inplus \mathbb{C},  ~ D_5 \inplus V_{D_5}(\lambda_4), ~ \text{and} ~ D_5 \inplus V_{D_5}(\lambda_5).
\end{array}
\end{equation}
The highest weight vectors of $D_{5}$-invariant abelian subspaces of $E_6$ are as follows:
The only abelian subspace which is isomorphic to ${\Bbb C}$
with respect to the adjoint action of $\varphi_5(D_5)$ or $\varrho_5(D_5)$ has
(highest weight) vector $\alpha  H$, for $\alpha \in \mathbb{C}^*$.

The abelian subspaces of  $E_6$ which carry $V_{D_5}(\lambda_5)$ irreps of
$D_{5}$ with respect to the adjoint
action of $\varphi_5(D_5)$ or $\varrho_5(D_5)$  have highest weight vectors
$\alpha  X''$
or $\alpha  Y_1$, respectively, for $\alpha \in \mathbb{C}^*$.
The abelian subspaces of  $E_6$ which carry
$V_{D_5}(\lambda_4)$ irreps of
$D_{5}$ with respect to the adjoint
action of $\varphi_5(D_5)$ or $\varrho_5(D_5)$ have highest weight vectors
$\alpha  Y_1$
or $\alpha  X''$, respectively, for $\alpha \in \mathbb{C}^*$.
\end{proposition}

\section{Classifying the embeddings of the abelian extensions of  ${D}_n$ into $E_{n+1}$}\label{sec7}

In this section, we classify, up to inner automorphism,
 the embeddings identified in Section \ref{sec6} of each abelian extension of $D_n$ that may be embedded into $E_{n+1}$.  The classifications are  summarized in Theorems \ref{classaa}, \ref{classaab}, and \ref{classaaddd}.

\subsection{Embeddings of the abelian extensions of  ${D}_7$ into $E_{8}$}

We begin by constructing lifts of the natural embedding $\varphi_7$  to $D_7 \inplus \mathbb{C}^{14}$, and to $D_7 \inplus \mathbb{C}^{}$.
Let $u$ be a highest weight vector of the $D_7$-representation $V_{D_7}(\lambda_1)\cong \mathbb{C}^{14}$.  Then, a lift of $\varphi_7$ to $D_7 \inplus \mathbb{C}^{14}$
is uniquely determined by its evaluation on $u$.  By Proposition \ref{mhh}, the following then define all possible lifts of the  embedding $\varphi_7$ to $D_7 \inplus \mathbb{C}^{14}$  for each $\alpha \in \mathbb{C}^*$:
\begin{equation}\label{jh}
\begin{array}{cccccccccccccccc}
\begin{array}{cccccccccccc}
\widetilde{\varphi_7}^{\lambda_1,  \alpha}: &D_7 \inplus \mathbb{C}^{14} &\hookrightarrow & E_8\\
&u & \mapsto & \alpha X''', \\
\widetilde{\varphi_7}^{\lambda'_1,  \alpha}: &D_7 \inplus \mathbb{C}^{14} &\hookrightarrow & E_8\\
&u & \mapsto & \alpha Y'.
\end{array}
\end{array}
\end{equation}
Similarly, the following defines  all possible lifts of the  embedding $\varphi_7$ to $D_7 \inplus \mathbb{C}^{}$  for each $\alpha \in \mathbb{C}^*$.
Here $u$ is any nonzero vector in $V_{D_7}(0) \cong \mathbb{C}$,  necessarily
of highest weight.
\begin{equation}\label{jhg}
\begin{array}{cccccccccc}
\widetilde{\varphi_7}^{0, \alpha}: &D_7 \inplus \mathbb{C}^{} &\hookrightarrow & E_8\\
&u & \mapsto & \alpha H.
\end{array}
\end{equation}
The embeddings of Eqs. \eqref{jh} and \eqref{jhg} form a basis for the classification of abelian extension of $D_7$ into $E_8$, up to equivalence, described in Theorem \ref{classaa}.  We first present three useful lemmas.

\begin{lemma}\label{pa}
The embeddings $\widetilde{\varphi_7}^{\lambda_1, 1}$ and  $\widetilde{\varphi_7}^{\lambda'_1, 1}$ are not equivalent embeddings of $D_7 \inplus \mathbb{C}^{14}$ into $E_8$.
\end{lemma}
\begin{proof}
By way of contradiction, suppose $\widetilde{\varphi_7}^{\lambda_1,1} \sim \widetilde{\varphi_7}^{\lambda'_1,1}$.    Let $\rho: E_8 \rightarrow E_8$ be an inner automorphism of $E_8$ such that
$\rho \circ \widetilde{\varphi_7}^{\lambda_1,1} = \widetilde{\varphi_7}^{\lambda'_1,1}$, which implies
\begin{equation}\label{pqa}
\begin{array}{llllllllll}
\rho(X''')&=& Y'.
\end{array}
\end{equation}
The automorphism $\rho$ must send highest weight vectors to highest weight vectors of equal weight with respect to the adjoint action of $\varphi(D_7)$, hence
\begin{equation}\label{kcx}
\begin{array}{llllllllll}
\rho(X'')&=& \alpha X'', & \rho(H) = \beta H,
\end{array}
\end{equation}
for
some
$\alpha, \beta \in \mathbb{C}^*$.  Eq. \eqref{kcx} yields
\begin{equation}
\begin{array}{lllllll}
\displaystyle \rho([H,X''])&=&\rho(X'')&=&\alpha X'',\\
\displaystyle [\rho(H),\rho(X'')]&=& \alpha \beta X''.
\end{array}
\end{equation}
Hence, $\beta=1$ so that $\rho(H)=H$.  Eq. \eqref{pqa} then yields
\begin{equation}
\begin{array}{lllllll}
\displaystyle \rho([H,X'''])&=&\rho(2X''')&=&2Y',\\
\displaystyle [\rho(H),\rho(X''')]&=& [H,Y']&=&-2Y',
\end{array}
\end{equation}
a contradiction to $\rho$ being a Lie algebra homomorphism.
  \end{proof}

\begin{lemma}\label{paa}
The embeddings $\widetilde{\varphi_7}^{\lambda_1,1}$ and  $\widetilde{\varphi_7}^{\lambda_1,\alpha}$  are equivalent for all $\alpha \in \mathbb{C}^*$.  The embeddings $\widetilde{\varphi_7}^{\lambda'_1,1}$ and  $\widetilde{\varphi_7}^{\lambda'_1,\alpha}$  are equivalent for all $\alpha \in \mathbb{C}^*$.
\end{lemma}
\begin{proof}
 For $\beta \in \mathbb{C}^*$, define an inner automorphism $\rho_\beta:E_8 \rightarrow E_8$ as follows:
\begin{equation}
\begin{array}{llllllllllll}
& H_{1} & \mapsto & H_{1}, & X_{1} & \mapsto & \beta X_{1},& Y_{1} & \mapsto & \frac{1}{\beta}Y_{1},\\
& H_{i} & \mapsto & H_{i}, & X_{i} & \mapsto &  X_{i},& Y_{i} & \mapsto & Y_{i},\\
\end{array}
\end{equation}
where $2\leq i\leq 8$.  We then have $\rho_{\beta} \circ \widetilde{\varphi_7}^{\lambda_1,1}=\widetilde{\varphi_7}^{\lambda_1,\alpha}$, $\rho_{\beta} \circ \widetilde{\varphi_7}^{\lambda'_1,\alpha}=\widetilde{\varphi_7}^{\lambda'_1,1}$, for $\beta^2 = \alpha$.  Hence $\widetilde{\varphi_7}^{\lambda_1,1}\sim \widetilde{\varphi_7}^{\lambda_1,\alpha}$ and $\widetilde{\varphi_7}^{\lambda'_1,1}\sim \widetilde{\varphi_7}^{\lambda'_1,\alpha}$.
\end{proof}

\begin{lemma}\label{paaa}
The embeddings $\widetilde{\varphi_7}^{0, \alpha}$ and  $\widetilde{\varphi_7}^{0, \beta}$ are equivalent if and only if $\alpha=\beta$.
\end{lemma}
\begin{proof}
Let $\widetilde{\varphi_7}^{0, \alpha} \sim \widetilde{\varphi_7}^{0, \beta}$, and let $\rho$ be an automorphism of
$E_8$ such that $\rho \circ \widetilde{\varphi_7}^{0, \alpha} = \widetilde{\varphi_7}^{0, \beta}$, which implies
\begin{equation}\label{zz1}
\begin{array}{lll}
\rho(H) = \frac{\beta}{\alpha} H=& \\
\frac{\beta}{\alpha}(4H_1+5 H_2+ 7H_3 +10H_4 +
 8H_5  + 6H_6
 +4H_7 +  2H_8).
 \end{array}
\end{equation}
Since $\rho$ is a Lie algebra homomorphism, and $\rho(H_i)=H_i$ for $2 \leq i \leq 8$, we also have
\begin{equation}\label{zz2}
\begin{array}{lllll}
\rho(H) = &\\4 \rho(H_1)+5 H_2+ 7H_3 +10H_4 +
 8H_5  + 6H_6
 +4H_7 +  2H_8.
\end{array}
\end{equation}
Eqs. \eqref{zz1} and \eqref{zz2} then yield
\begin{equation}\label{zz3}
\begin{array}{llllllllll}
\rho(H_1)= \frac{\beta}{\alpha} H_1 + \\ \frac{(\frac{\beta}{\alpha} -1)}{4}(5 H_2+ 7H_3 +10H_4 +
 8H_5  + 6H_6
 +4H_7 +  2H_8).
 \end{array}
\end{equation}
The automorphism $\rho$ must send   highest weight vectors, with respect to the adjoint action of $\varphi_7(D_7)$,  to highest weight vectors of the same weight.    Hence, there is a $\gamma \in \mathbb{C}^*$ such that
\begin{equation}\label{zz4}
\rho(Y_1) = \gamma Y_1.
\end{equation}
Using Eq. \eqref{zz3},
\begin{equation}\label{zz5}
\begin{array}{llllllllll}
\displaystyle \rho( [H_1, Y_1])&=&\rho( -2Y_1)&=& -2\gamma Y_1,\\
 \displaystyle [\rho(H_1),\rho(Y_1)  ]&=& \gamma [\rho(H_1),Y_1]&= &-\frac{\gamma}{4}(\frac{\beta}{\alpha}+7)Y_1.
\end{array}
\end{equation}
Since $\rho$ is a Lie algebra homomorphism, Eq. \eqref{zz5} implies $\frac{\beta}{\alpha}=1$.  Hence, if $\widetilde{\varphi_7}^{0, \alpha} \sim \widetilde{\varphi_7}^{0, \beta}$, then $\alpha=\beta$.  The opposite implication is obvious.
\end{proof}

\begin{theorem}\label{classaa}
The only abelian extensions of $D_7$ that may be embedded into $E_8$ are
$
D_7 \inplus V_{D_{7}}(\lambda_{1}) \cong
D_7 \inplus \mathbb{C}^{14}$
and
$
D_7 \inplus V_{D_{7}}(0) \cong
D_7 \inplus \mathbb{C}^{}$.
The embeddings $\widetilde{\varphi_7}^{\lambda_1, 1}$ and $\widetilde{\varphi_7}^{\lambda'_1,1}$ are a complete set of inequivalent embeddings of $D_7 \inplus \mathbb{C}^{14}$ into $E_8$.
There is an infinite family of inequivalent embeddings of $D_7 \inplus \mathbb{C}$ into $E_8$, which contains all embeddings.  The infinite family is parameterized by a single continuous parameter:  $\widetilde{\varphi_7}^{0, \alpha}$, $\alpha \in \mathbb{C}^*$.  These results are displayed in Table \ref{end}.
\end{theorem}
\begin{proof}
The only abelian extensions of $D_7$ that may be embedded into $E_8$ are $D_7 \inplus \mathbb{C}^{14}$, and $D_7 \inplus \mathbb{C}^{}$ (a repetition of Proposition \ref{mhh}).

Any embedding of an abelian extension of $D_7$ into $E_8$ restricted to $D_7$ is equivalent to $\varphi_7$  by Theorem \ref{dymin}.  Further, Eqs. \eqref{jh} and \eqref{jhg} list all possible lifts of $\varphi_7$  to the permissible abelian extensions of $D_7$.  The result then follows from Lemmas \ref{pa}, \ref{paa}, and \ref{paaa}.
\end{proof}

\begin{table} [!h]\renewcommand{\arraystretch}{1.6} \caption{Classification of embeddings of abelian extensions of $D_7$ into $E_8$ up to equivalence. } \label{end}\begin{center}
\begin{tabular}{|c|c|c|clclcl}
\hline
Abelian Extension $D_7 \inplus V$ &  Embedding $D_7 \inplus V \hookrightarrow E_8$\\
\hline \hline
$
D_7 \inplus V_{D_{7}}(\lambda_{1}) \cong
D_7 \inplus \mathbb{C}^{14}$& $\widetilde{\varphi_7}^{\lambda_1,1}$, ~~   $\widetilde{\varphi_7}^{\lambda'_1, 1}$\\
\hline
$
D_7 \inplus V_{D_{7}}(0) \cong
D_7 \inplus \mathbb{C}^{}$ &  $\widetilde{\varphi_7}^{0, \alpha}$, ~~ $\alpha \in \mathbb{C}^*$\\
\hline
\end{tabular}\end{center}
\end{table}

\begin{remark}
The Euclidean algebra $\mathfrak{e}(n)$ of isometries of $n$-dimensional Euclidean space is defined as
$\mathfrak{e}(n) \cong \mathfrak{so}(n,\mathbb{C}) \inplus \mathbb{C}^n$.
We thus have
$\mathfrak{e}(14) \cong D_7 \inplus \mathbb{C}^{14}$.
Hence, one consequence of Theorem \ref{classaa}  is a classification of the embeddings of the Euclidean algebra $\mathfrak{e}(14)$ into $E_8$, up to equivalence.
\end{remark}

\subsection{Embeddings of the abelian extensions of  ${D}_6$ into $E_{7}$}

We begin by constructing lifts of the embeddings $\varphi_6: D_6 \hookrightarrow E_7$ and $\varrho_6: D_6 \hookrightarrow E_7$
to $
D_6 \inplus V_{D_{6}}(0) \cong
D_6 \inplus \mathbb{C}^{}$.  Let $u$ be a
(highest weight)
vector of the $D_6$-representation $V_{D_6}(0)\cong \mathbb{C}^{}$.  By Proposition \ref{mhhh}, the following then define all possible lifts of the  embeddings $\varphi_6$ and $\varrho_6$  to $D_6 \inplus \mathbb{C}^{}$,  for  $\alpha, \beta, \gamma \in \mathbb{C}$, not all zero:

\begin{equation}\label{ggyy}
\begin{array}{lll}
\begin{array}{ccccccccccc}
\widetilde{\varphi_6}^{0,(\alpha,\beta, \gamma)}: &D_6 \inplus \mathbb{C}^{} &\hookrightarrow & E_7\\
&u & \mapsto &  \alpha Y' + \beta X''' + \gamma H,
\end{array}\\
\begin{array}{cccccccccccc}
\widetilde{\varrho_6}^{0,(\alpha,\beta, \gamma)}: &D_6 \inplus \mathbb{C}^{} &\hookrightarrow & E_7\\
&u & \mapsto &  \alpha Y' + \beta X''' + \gamma H.
\end{array}
\end{array}
\end{equation}

The embeddings of Eq. \eqref{ggyy} form a basis for the classification of abelian
extensions
of $D_6$ into $E_7$, up to equivalence, described in Theorem \ref{classaab}.  We first present a useful lemma.

\begin{lemma}\label{pj2}
Let $\rho$ be an automorphism of $E_7$  which fixes $X_i, Y_i$, and $H_i$ for $2 \leq i \leq 7$.  Then
\begin{equation}
\begin{array}{lllllllllll}
\rho(Y') &=& c^2Y'-cdH-d^2X''', \\
\rho(X''') &=& a^2X'''+abH-b^2Y',\\
 \rho(H)&=&(ac+bd)H+2adX'''-2bcY',
\end{array}
\end{equation}
where $a, b, c, d \in \mathbb{C}$ such that $ac-bd=1$.
\end{lemma}
\begin{proof}
Let $\rho: E_7 \rightarrow E_7$ be an automorphism of $E_7$ which  fixes $X_i, Y_i$, and $H_i$ for $2 \leq i \leq 7$.
A highest weight vector of $E_7$ must be sent to a highest weight vector of $E_7$ of equal weight with respect to the adjoint action of $\varphi_6(D_6)$.  Hence, referring to Eq. \eqref{hhhh},
\begin{equation}\label{u3}
\begin{array}{llllllll}
\rho(Y_1) &=& c Y_1 + d X'',  &
\rho(X'') &=& a X'' +b Y_1.
\end{array}
\end{equation}
Computation shows that there exists a sequence $a_1, a_2,...,a_k$, where $2\leq a_i \leq 7$ for each $i$, such that
\begin{equation}\label{gfd}
\begin{array}{lllll}
\displaystyle [[[[[X'', Y_{a_1}],Y_{a_2}],Y_{a_3}]\cdots], Y_{a_k}]  &=&-X_1, ~\text{and}\\
\displaystyle  [[[[[Y_1, Y_{a_1}],Y_{a_2}],Y_{a_3}]\cdots], Y_{a_k}]  &=&-Y_{7,6,5,4,3,2,4,5,6,1,3,4,5,2,4,3}.
 \end{array}
\end{equation}
Let us define $Y''= Y_{7,6,5,4,3,2,4,5,6,1,3,4,5,2,4,3}$.   Note that the elements $X_1$ and $Y''$ are lowest weight vectors with respect to the adjoint action of $\varphi_6(D_6)$.  Equations  \eqref{u3} and \eqref{gfd}, together with the fact that $\rho(Y_{a_i})=Y_{a_i}$ for $2\leq a_i \leq 7$,  then imply
\begin{equation}\label{u4}
\begin{array}{llllllllll}
\rho(Y'') &=& c Y'' +d X_1, &
\rho(X_1) &=& aX_1 +b Y''.
\end{array}
\end{equation}
Note that $[X_1,X'']=X'''$ and  $[Y_1,Y'']=Y'$, so that Eqs. \eqref{u3} and \eqref{u4} imply
\begin{equation}\label{ut4}
\begin{array}{llllllllll}
\rho(X''') &=& a^2X'''+abH-b^2Y', \\
\rho(Y') &=& c^2Y'-cdH-d^2X'''.
\end{array}
\end{equation}
Note that
\begin{equation}\label{asxt555}
\begin{array}{lllll}
\displaystyle [\rho(Y_1), \rho(X''')]&=&[cY_1+dX'',a^2X'''+abH-b^2Y']\\
\displaystyle &=&b(ac-bd)Y_1+a(ac-bd)X''.
\end{array}
\end{equation}
Since $[Y_1,X''']=X''$, Eqs. \eqref{u3} and \eqref{asxt555} imply $b(ac-bd)=b$ and $a(ac-bd)=a$.  Since not both $a=0$ and $b=0$, we have
\begin{equation}
ac-bd=1.
\end{equation}
Since $[X''',Y']=H$ and $ac-bd=1$, Eq. \eqref{ut4} implies
\begin{equation}\label{asxt55}
\begin{array}{lllll}
\displaystyle \rho(H)=(ac+bd)H+2adX'''-2bcY'.
\end{array}
\end{equation}
\end{proof}

\begin{theorem}\label{classaab}
The only abelian extension of $D_6$ that may be embedded into $E_7$ is
$
D_6 \inplus V_{D_{6}}(0)  \cong
D_6 \inplus \mathbb{C}^{}$.
Any embedding of $D_6 \inplus \mathbb{C}$ into $E_7$ is equivalent to $\widetilde{\varphi_6}^{0,(\alpha,\beta, \gamma)}$ or $\widetilde{\varrho_6}^{0,(\alpha,\beta, \gamma)}$ for some $\alpha, \beta, \gamma \in \mathbb{C}$, not all zero.  For $\alpha, \beta, \gamma \in \mathbb{C}$, not all zero, the embeddings are classified according to the following rules:

\begin{enumerate}[(a)]
\item  $\widetilde{\varphi_6}^{0,(\alpha,\beta, \gamma)} \nsim \widetilde{\varrho_6}^{0,(\alpha,\beta, \gamma)} ~\text{for all}~~ \alpha, \beta, \gamma$.
\item  $\widetilde{\varphi_6}^{0,(\alpha,\beta, \gamma)} \sim \widetilde{\varphi_6}^{0,(\alpha',\beta', \gamma')}\Leftrightarrow$\\
$  (\alpha', \beta', \gamma' ) =
 (\alpha  c^2-\beta b^2-2\gamma bc, -\alpha d^2+\beta a^2+2\gamma ad, -\alpha cd +\beta ab+\gamma (ac+bd)), ~\text{for some}~ a,b,c,d \in \mathbb{C},~\text{such that}~ ac-bd=1$.
\item $\widetilde{\varrho_6}^{0,(\alpha,\beta, \gamma)} \sim \widetilde{\varrho_6}^{0,(\alpha',\beta', \gamma')}\Leftrightarrow$ \\
$  (\alpha', \beta', \gamma' ) =
 (\alpha  c^2-\beta b^2-2\gamma bc, -\alpha d^2+\beta a^2+2\gamma ad, -\alpha cd +\beta ab+\gamma (ac+bd)), ~\text{for some}~ a,b,c,d \in \mathbb{C},~\text{such that}~ ac-bd=1$.
\end{enumerate}
\end{theorem}
\begin{proof}
Let $\widetilde{\varphi_6}^{0,(\alpha,\beta, \gamma)} \sim \widetilde{\varphi_6}^{0,(\alpha',\beta', \gamma')}$, and let $\rho$ be an automorphism of $E_7$ such that
\begin{equation}\label{u10}
\rho \circ \widetilde{\varphi_6}^{0,(\alpha,\beta, \gamma)} = \widetilde{\varphi_6}^{0,(\alpha',\beta', \gamma')}.
\end{equation}
Eq. \eqref{u10} implies
\begin{equation}\label{u12}
\rho(\alpha Y' + \beta X''' + \gamma H ) = \alpha' Y' + \beta' X''' + \gamma' H.
\end{equation}
Eq. \eqref{u10} implies that $\rho$ fixes $X_i, Y_i$, and $H_i$ for $2 \leq i \leq 7$.  Hence Lemma \ref{pj2} implies
\begin{equation}\label{u14b}
\begin{array}{llllllll}
 \rho(\alpha Y' + \beta X''' + \gamma H ) = \alpha\rho(Y') + \beta\rho(X''') + \gamma \rho(H)=\\ (\alpha  c^2-\beta b^2-2\gamma bc) Y'
 +(-\alpha d^2+\beta a^2+2\gamma ad)X'''+\\( -\alpha cd +\beta ab+\gamma (ac+bd))H,
\end{array}
\end{equation}
for some $a, b, c, d \in \mathbb{C}$ such that $ac-bd=1$.  Hence, Eqs. \eqref{u12} and \eqref{u14b} imply $(\alpha', \beta', \gamma' ) =$  $(\alpha  c^2-\beta b^2-2\gamma bc, -\alpha d^2+\beta a^2+2\gamma ad, -\alpha cd +\beta ab+\gamma (ac+bd))$.

We now prove the opposite implication of item (b).  Let $\rho: E_7 \rightarrow E_7$ be the (inner) automorphism of $E_7$ that fixes $X_i, Y_i$, and $H_i$ for $2 \leq i \leq 7$, and
\begin{equation}\label{u14c}
\begin{array}{llllllll}
\rho(X_1)&=& aX_1+bY'',\\
\rho(Y_1)&=& cY_1+dX'',\\
\rho(H_1)&=& H_1+bdH+adX'''-bcY'.
\end{array}
\end{equation}
Then,
\begin{equation}
\begin{array}{lllllllllll}
\rho(Y') &=& c^2Y'-cdH-d^2X''', \\
\rho(X''') &=& a^2X'''+abH-b^2Y',\\
 \rho(H)&=&(ac+bd)H+2adX'''-2bcY',
\end{array}
\end{equation}
so that
\begin{equation}\label{u14bbbb}
\begin{array}{llllllll}
 \rho(\alpha Y' + \beta X''' + \gamma H ) = \alpha' Y'+\beta'X'''+\gamma'H,
\end{array}
\end{equation}
as in Eq. \eqref{u14b}, which implies $\rho \circ \widetilde{\varphi_6}^{0,(\alpha,\beta, \gamma)} = \widetilde{\varphi_6}^{0,(\alpha',\beta', \gamma')}$.  Hence, we have established item (b).  Item (c) is proved in a similar manner.

Now we consider item (a).  Since $\varphi_6 \nsim \varrho_6$ by Theorem \ref{dymin}, then  $\widetilde{\varphi_6}^{0,(\alpha,\beta, \gamma)}$ $\nsim$ $\widetilde{\varrho_6}^{0,(\alpha',\beta', \gamma')}$.
\end{proof}

\subsection{Embeddings of the abelian extensions of  ${D}_5$ into $E_{6}$}

We begin by constructing lifts of the embeddings $\varphi_5$ and $\varrho_5$  to $D_5 \inplus V_{D_5}(\lambda_5)$, $D_5 \inplus V_{D_5}(\lambda_4)$, and to
$
D_5 \inplus V_{D_{5}}(0) \cong
D_5 \inplus \mathbb{C}^{}$.
Let $u$ be a highest weight vector of the $D_5$-representation $V_{D_5}(\lambda_5)$, $V_{D_5}(\lambda_4)$,  or $\mathbb{C}$, respectively.  By Proposition \ref{mhhhh}, the following then define all possible lifts of  $\varphi_5$ or $\varrho_5$ to $D_5 \inplus V_{D_5}(\lambda_5)$, $D_5 \inplus V_{D_5}(\lambda_4)$, and to $D_5 \inplus \mathbb{C}^{}$, respectively, for each $\alpha \in \mathbb{C}^*$:
\begin{equation}\label{qqwwe}
\begin{array}{llllllllllllllllll}
\begin{array}{ccccccccccccc}
\widetilde{\varphi_5}^{\lambda_5, \alpha}: &D_5 \inplus V_{D_5}(\lambda_5) &\hookrightarrow & E_6\\
&u & \mapsto & \alpha X'',&
\\
\widetilde{\varrho_5}^{\lambda_5, \alpha}: &D_5 \inplus V_{D_5}(\lambda_5) &\hookrightarrow & E_6\\
&u & \mapsto & \alpha  Y_1,&
\\
\widetilde{\varphi_5}^{\lambda_4, \alpha}: &D_5 \inplus V_{D_5}(\lambda_4) &\hookrightarrow & E_6\\
&u & \mapsto & \alpha Y_1,&
\\
\widetilde{\varrho_5}^{\lambda_4, \alpha}: &D_5 \inplus V_{D_5}(\lambda_4) &\hookrightarrow & E_6\\
&u & \mapsto & \alpha X'',&
\\
\widetilde{\varphi_5}^{0, \alpha}: &D_5 \inplus \mathbb{C} &\hookrightarrow & E_6\\
&u & \mapsto & \alpha H,&
\\
\widetilde{\varrho_5}^{0, \alpha}: &D_5 \inplus \mathbb{C} &\hookrightarrow & E_6\\
&u & \mapsto & \alpha H.&
\end{array}
\end{array}
\end{equation}
The embeddings of  \eqref{qqwwe} form the basis of the classification of abelian extension of $D_5$ into $E_6$, up to equivalence, in Theorem \ref{classaaddd}.  We first prove two useful lemmas.

\begin{lemma}\label{paacc}
For all $\alpha \in \mathbb{C}^*$,
\begin{equation}
\begin{array}{lllllllllll}
\widetilde{\varphi_5}^{\lambda_5,1}&\sim&\widetilde{\varphi_5}^{\lambda_5,\alpha}, &  \widetilde{\varrho_5}^{\lambda_5,1}&\sim&\widetilde{\varrho_5}^{\lambda_5,\alpha},  \\
 \widetilde{\varphi_5}^{\lambda_4,1}&\sim&\widetilde{\varphi_5}^{\lambda_4,\alpha}, &
  \widetilde{\varrho_5}^{\lambda_4,1}&\sim&\widetilde{\varrho_5}^{\lambda_4,\alpha},\\
    \widetilde{\varphi_5}^{\lambda_4,1}&\nsim&\widetilde{\varrho_5}^{\lambda_4,1}, &  \widetilde{\varphi_5}^{\lambda_5,1}&\nsim&\widetilde{\varrho_5}^{\lambda_5,1}.\\
\end{array}
\end{equation}
\end{lemma}
\begin{proof}
 Define an inner automorphism  $\rho:E_6 \rightarrow E_6$ as follows:
\begin{equation}
\begin{array}{llllllllllll}
& H_{1} & \mapsto & H_{1},& X_{1} & \mapsto & \alpha X_{1}, & Y_{1} & \mapsto & \frac{1}{\alpha} Y_{1}, \\
& H_{i} & \mapsto & H_{i},& X_{i} & \mapsto &  X_{i}, & Y_{i} & \mapsto &  Y_{i},
\end{array}
\end{equation}
where $2\leq i\leq 6$.  We then have
\begin{equation}
\begin{array}{lllllllllllll}
\rho_{} \circ \widetilde{\varphi_5}^{\lambda_5,1}&=&\widetilde{\varphi_5}^{\lambda_5,\alpha}, & \rho_{} \circ \widetilde{\varrho_5}^{\lambda_5,\alpha}&=&\widetilde{\varrho_5}^{\lambda_5,1}\\
\rho_{} \circ \widetilde{\varphi_5}^{\lambda_4,\alpha}&=&\widetilde{\varphi_5}^{\lambda_4,1}, & \rho_{} \circ \widetilde{\varrho_5}^{\lambda_4,1}&=&\widetilde{\varrho_5}^{\lambda_4,\alpha}.\\
\end{array}
\end{equation}
Since $\varphi_5 \nsim \varrho_5$ by Theorem \ref{dymin}, then $\widetilde{\varphi_5}^{\lambda_4,1}\nsim\widetilde{\varrho_5}^{\lambda_4,1}$, and
  $\widetilde{\varphi_5}^{\lambda_5,1}\nsim\widetilde{\varrho_5}^{\lambda_5,1}$. \end{proof}

\begin{lemma}\label{paaccc}
  The embeddings $\widetilde{\varphi_5}^{0, \alpha}$ and  $\widetilde{\varphi_5}^{0, \beta}$  are equivalent if and only if $\alpha = \beta$.    Similarly, the embeddings $\widetilde{\varrho_5}^{0, \alpha}$ and  $\widetilde{\varrho_5}^{0, \beta}$  are equivalent if and only if $\alpha = \beta$.   Finally,  $\widetilde{\varphi_5}^{0, \alpha}\nsim \widetilde{\varrho_5}^{0, \beta}$, for all $\alpha, \beta$.
\end{lemma}
\begin{proof}
Let $\widetilde{\varphi_5}^{0, \alpha} \sim \widetilde{\varphi_5}^{0, \beta}$, and let $\rho$ be an automorphism of
$E_6$ such that $\rho \circ \widetilde{\varphi_5}^{0, \alpha} = \widetilde{\varphi_5}^{0, \beta}$, which implies
\begin{equation}\label{z1}
\rho(H) = \frac{\beta}{\alpha} H= \frac{\beta}{\alpha}
\biggl(
2H_1+\frac{3}{2}H_2+\frac{5}{2}H_3+3H_4+2H_5+H_6
\biggr).
\end{equation}
Since $\rho$ is a Lie algebra homomorphism, and $\rho(H_i)=H_i$ for $2 \leq i \leq 6$, we also have
\begin{equation}\label{z2}
\rho(H) = 2 \rho(H_1)+\frac{3}{2}H_2+\frac{5}{2}H_3+3H_4+2H_5+H_6 .
\end{equation}
Eqs. \eqref{z1} and \eqref{z2} then yield
\begin{equation}\label{z3}
\rho(H_1) = \frac{\beta}{\alpha} H_1+
\biggl(
\frac{\beta}{\alpha}-1
\biggr)
\biggl(
\frac{3}{4}H_2+\frac{5}{4}H_3+\frac{3}{2}H_4+H_5+\frac{1}{2}H_6
\biggr).
\end{equation}
The automorphism must send  highest weight vectors, with respect to the adjoint action of $\varphi_5(D_5)$,  to highest weight vectors of the same weight.  Hence, for some $\gamma \in \mathbb{C}^*$,
\begin{equation}\label{z4}
\rho(Y_1) = \gamma Y_1.
\end{equation}
Considering Eqs. \eqref{z3} and \eqref{z4},
\begin{equation}\label{z5}
\begin{array}{llllllllll}
\displaystyle \rho( [H_1, Y_1])&=&\rho( -2Y_1)=-2 \gamma Y_1,\\
 \displaystyle [\rho(H_1),\rho(Y_1)  ]&=& -\frac{\gamma}{4} (3\frac{\beta}{\alpha}+5)Y_1.
\end{array}
\end{equation}
Since $\rho$ is a Lie algebra homomorphism, Eq. \eqref{z5} implies $\frac{\beta}{\alpha}=1$.  Hence, if $\widetilde{\varphi_5}^{0, \alpha} \sim \widetilde{\varphi_5}^{0, \beta}$, then $\alpha=\beta$.  The opposite implication is obvious.

Similarly we prove $\widetilde{\varrho_5}^{0, \alpha} \sim \widetilde{\varrho_5}^{0, \beta}$ if and only if $\alpha = \beta$.
 Finally, since $\varphi_5 \nsim \varrho_5$ by Theorem \ref{dymin}, then $\widetilde{\varphi_5}^{0,\alpha}\nsim\widetilde{\varrho_5}^{0,\beta}$ for all $\alpha, \beta$.
\end{proof}

\begin{theorem}\label{classaaddd}
The only abelian extensions of $D_5$ that may be embedded into $E_6$ are $D_5 \inplus V_{D_7}(\lambda_5)$, $D_5 \inplus V_{D_7}(\lambda_4)$, and $D_5 \inplus \mathbb{C}^{}$.  The embeddings $\widetilde{\varphi_5}^{\lambda_5, 1}$ and $\widetilde{\varrho_5}^{\lambda_5,1}$ are a complete set of inequivalent embeddings of $D_5 \inplus V_{D_7}(\lambda_5)$ into $E_6$.  The embeddings $\widetilde{\varphi_5}^{\lambda_4, 1}$ and $\widetilde{\varrho_5}^{\lambda_4,1}$ are a complete set of inequivalent embeddings of $D_5 \inplus V_{D_7}(\lambda_4)$ into $E_6$.
There are two infinite families of inequivalent embeddings of $D_5 \inplus \mathbb{C}$ into $E_6$, which contain all
inequivalent embeddings.  Each family is parameterized by a single continuous parameter:   $\widetilde{\varphi_5}^{0, \alpha}$ and $\widetilde{\varrho_5}^{0,\alpha}$, for $\alpha \in \mathbb{C}^*$.   These results are displayed in Table \ref{enddd}.
\end{theorem}
\begin{proof}
The only abelian extensions of $D_5$ that may be embedded into $E_6$ are $D_5 \inplus V_{D_7}(\lambda_5)$, $D_5 \inplus V_{D_7}(\lambda_4)$, and $D_5 \inplus \mathbb{C}^{}$ (a repetition of Proposition \ref{mhhhh}).

The restriction to $D_5$ of any embedding of an abelian extension of $D_5$ into $E_6$  is equivalent to $\varphi_5$ or $\varrho_5$ by Theorem \ref{dymin}.  Further, \eqref{qqwwe} lists all possible lifts of $\varphi_5$ or $\varrho_5$ to the permissible abelian extensions of $D_5$.  The result then follows from Lemmas \ref{paacc} and \ref{paaccc}.
\end{proof}

\begin{table} [!h]\renewcommand{\arraystretch}{1.6} \caption{Classification of embeddings of abelian extensions of $D_5$ into $E_6$ up to equivalence.} \label{enddd}\begin{center}
\begin{tabular}{|c|c|c|clclcl}
\hline
Abelian Extension $D_5 \inplus V$ &  Embedding $D_5 \inplus V \hookrightarrow E_6$\\
\hline \hline
$D_5 \inplus V_{D_5}(\lambda_5)$& $\widetilde{\varphi_5}^{\lambda_5, 1}$, ~~$\widetilde{\varrho_5}^{\lambda_5, 1}$\\
\hline
$D_5 \inplus V_{D_5}(\lambda_4)$& $\widetilde{\varphi_5}^{\lambda_4, 1}$, ~~$\widetilde{\varrho_5}^{\lambda_4, 1}$ \\
\hline
$
D_5 \inplus V_{D_{5}}(0) \cong
D_5 \inplus \mathbb{C}^{}$ &  $\widetilde{\varphi_5}^{0, \alpha}$, ~~$\widetilde{\varrho_5}^{0, \alpha}$, ~~ $\alpha \in \mathbb{C}^*$\\
\hline
\end{tabular}\end{center}
\end{table}

\section{Representations of  $D_{n}\inplus V$ from representations of $E_{n+1}$}\label{sec8}

As an application of the above classification, we also consider the irreducible representations of $E_{n+1}$ restricted to $D_n \inplus V$, for each abelian extension and each embedding.  The results are summarized in Theorem \ref{ind} and Examples \ref{indaa} to \ref{indc} below.

\begin{theorem}\label{ind}
Any irreducible representation of $E_{6}$ restricts to an indecomposable representation of $D_{5}\inplus V_{D_5}(\lambda_4)$ or $D_{5}\inplus V_{D_5}(\lambda_5)$ with respect to any embedding of $D_{5}\inplus V_{D_5}(\lambda_4)$ or $D_{5}\inplus V_{D_5}(\lambda_5)$  into $E_6$, respectively.
\end{theorem}

\begin{proof}
The  theorem follows immediately from the observation that the image of $D_5 \inplus V_{D_5}(\lambda_4)$ or $D_5 \inplus V_{D_5}(\lambda_5)$ under any embedding into $E_6$, all of which were identified in Theorem \ref{classaaddd}, contains
all positive or  all negative root vectors of $E_6$ (see for instance  \cite{dr3});
thus its proof is
omitted.
\end{proof}

 The following three examples show that for every case other than the two considered in Theorem \ref{ind}, an irreducible representation of
 $E_{n+1}$ may decompose when restricted to an abelian extension $D_n \inplus V$, under any embedding.

\begin{example}\label{indaa}
Irreducible representations of $E_7$ may decompose when restricted to $D_6 \inplus \mathbb{C}$.  For instance, direct calculation gives us the following decomposition of the $E_7$ fundamental representation  $V_{E_7}(\lambda_1)$:
\begin{equation}
\begin{array}{llllllllllllll}
V_{E_7}(\lambda_1)|_{\widetilde{\varphi_7}^{0,(\alpha,\beta, \gamma)}} &\cong&
 V_{D_6} (\lambda_2) \oplus (V_{D_6}(\lambda_5) + V_{D_6}(\lambda_5)) \oplus \\
 &&  (V_{D_6}(0) + V_{D_6}(0)+ V_{D_6}(0)),\\
V_{E_7}(\lambda_1)|_{\widetilde{\varrho_7}^{0,(\alpha,\beta, \gamma)}} &\cong&
 V_{D_6} (\lambda_2) \oplus (V_{D_6}(\lambda_6) + V_{D_6}(\lambda_6))\oplus\\
  && (V_{D_6}(0) + V_{D_6}(0)+ V_{D_6}(0)).\\
\end{array}
\end{equation}
The decomposition  of $V_{D_6}(\lambda_5) + V_{D_6}(\lambda_5)$, $V_{D_6}(\lambda_6) + V_{D_6}(\lambda_6)$, and
$V_{D_6}(0) + V_{D_6}(0)+ V_{D_6}(0)$  will depend on the values of $\alpha, \beta$, and $\gamma$.
\end{example}

\begin{example}\label{inda}
The adjoint representation of $E_6$  decomposes when restricted to $D_5 \inplus \mathbb{C}$, with respect to any embedding.  Similarly, the adjoint
representation
of $E_8$  decomposes when restricted to $D_7 \inplus \mathbb{C}$, with respect to any embedding.  In each case,  the decomposition with respect to $D_5 \inplus \mathbb{C}$ or $D_7 \inplus \mathbb{C}$ is the same as the decomposition with respect to the adjoint action of the image of $D_5$ or $D_7$, respectively.

The reason for this decomposition is that the highest weight vector of $\mathbb{C}$, with respect to the adjoint action of the image of $D_5$ or $D_7$, is an element of a Cartan subalgebra of $E_6$ or $E_8$, respectively (see \eqref{iiii} and \eqref{iiiib} or \eqref{hhhhj}, respectively).

More generally,  the decomposition of an $E_6$ or $E_8$ irrep with respect to $D_5 \inplus \mathbb{C}$ or $D_7 \inplus \mathbb{C}$, respectively, is the same as the decomposition with respect to the restricted action of the image of $D_5$ or $D_7$, respectively.

\end{example}

\begin{example}\label{indc}
 Irreducible representations of $E_8$ may decompose when restricted to $D_7 \inplus \mathbb{C}^{14}$.
 For instance, direct calculation gives us the following decomposition of the $E_8$ fundamental representation $V_{E_8}(\lambda_8)$:
\begin{equation}
\begin{array}{llllllllllllllllll}
V_{E_8}(\lambda_8)|_{\widetilde{\varphi_7}^{\lambda_1,1}} \cong V_{E_8}(\lambda_8)|_{\widetilde{\varphi_7}^{\lambda'_1,1}}
\cong\\( V_{D_7} (\lambda_2) + 2V_{D_7} (\lambda_1)+ V_{D_7}(0))
\oplus (V_{D_7} (\lambda_6) + V_{D_7} (\lambda_7)).
\end{array}
\end{equation}
Each set of parentheses on the right side of the last equation contains a representation of ${D}_7\inplus \mathbb{C}^{14}$, described in terms of the decomposition of its restriction to $D_7$.
\end{example}

\section{Conclusions}

We determined all abelian extensions of $D_n$ that may be embedded into the exceptional Lie algebra $E_{n+1}$, $n=5, 6$, and $7$.  We then classified each of these embeddings, up to equivalence.  The classifications are summarized in Theorems \ref{classaa}, \ref{classaab}, and \ref{classaaddd}.

As an application of the  classification, we also considered the irreducible representations of $E_{n+1}$ restricted to $D_n \inplus V$, for each abelian extension and each embedding.  Any irreducible representation of $E_{6}$ restricts to an indecomposable representation of $D_{5}\inplus V_{D_5}(\lambda_4)$ or $D_{5}\inplus V_{D_5}(\lambda_5)$ with respect to any embedding.  For every other case, an irreducible representation of $E_{n+1}$ restricted to $D_n \inplus V$ may decompose.

\section*{Acknowledgements}

The work of A.D. and D.K. is partially supported by research grants from the Professional Staff Congress/City University of New York.  The work
of J.R. is partially supported by the Natural Sciences and Engineering Research Council.
We thank Andrey Minchenko for a helpful conversation on embeddings of simple Lie algebras into exceptional Lie algebras.

redpajama_set_name: RedPajamaArXiv


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If your Valentine appreciates good humor and an even better laugh, add a few funny Valentine's day quotes to your message. These quotes will help you make light of the holiday in good humor.
When you're trying to figure out what to write in a Valentine's day card for him just remember to keep your message personal. Before you begin writing your message, think about why this person is special to you. You can also think about things that the two of you do together and how they wouldn't be the same without him. You'd be surprised at how far a little personalization can go.
Use your pet name for him, jot down a list of inside jokes that only the two of you share and find some inspiration with these Valentines day quotes for him.
Remind her of the beauty you get to see everyday and fill the space of your Valentine's day card with words that will touch her heart. These Valentine's day quotes will elevate your message to match all of your special gift ideas for her this Valentine's day.
Friendship is an extraordinary kind of love that may not always get the credit it deserves. Whether your friendship is more playful or serious, sending a Valentine's day card to your friends is a great way to remind them how much you enjoy their company. Pair your card with one of our Valentine's day box ideas to surprise your friend with a thoughtful gift.
Be sure to remind those close to your heart what their friendship means to you with a few Valentine's day quotes.
Valentine's day is symbolic of all things adorable. From heart shaped candies to fluffy life-sized teddy bears and Valentine's cards, there may not be a more endearing holiday. This Valentine's day, make your Valentine feel darling and dedicate a quote to him or her. The only thing better than a Valentine's day message is a cute Valentine's day message, right?
Always remember to love yourself! These Valentine's day quotes serve as a little reminder of how important it is to love yourself first.
Don't forget to wish your Valentine a very happy Valentine's Day! We've listed below some of our favorite messages to wish your loved one well in the coming year. Pick your favorite or use them to inspire a special Valentine's wish of your very own.
Wishing you a Happy Valentine's Day spent with all the ones you love—especially me!
I wish you the most wonderful Valentine's Day—I know mine is better thanks to you!
Smarties: Who wouldn't love a smartie like you?
Chocolate: I love you a chocoLOT!
100 Grand Bar: Our friendship is worth a lot more than 100 Grand!
Gummy Bears: Wishing you a bear-y sweet Valentine's Day!
Candy String Necklace: I'm not just stringing you along! Happy Valentine's Day!
Other Candy: You make my Valentine's Day a whole lot sweeter!
Highlighter: You're the highlight to my day!
Marker: Take this as a mark of our friendship!
Pencil: You're one sharp Valentines, thanks for being you!
Eraser: Nothing could erase my love for you!
Stickers: I sure am stuck on you! Happy Valentine's Day!
Cookie Cutter: You're a cut above the rest, my friend!
Your time and recognition won't be forgotten by anyone who receives a heartfelt Valentine's wish from you. This year share the love of the holiday with everyone close to you. Valentine's day quotes serve as the perfect inspiration to help you get started on your messages and show those close to you just how much you care. Don't forget to sign your card with love and deliver with care (and Valentine's gifts!).


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General story • Opinion
For Native Americans, a river is more than a 'person.' It's also a sacred place
Horseshoe Bend, Colorado River. Photo by Paul Hermans, Wikimedia Creative Commons
Rosalyn R. LaPier
(The Conversation) The environmental group Deep Green Resistance recently filed a first-of-its-kind legal suit against the state of Colorado asking for personhood rights for the Colorado River.
If successful, it would mean lawsuits can brought on behalf of the river for any harm done to it, as if it were a person.
In the past, several environmental groups in India, Bolivia, Ecuador, Colombia and New Zealand have successfully sought protection for rivers and landscapes based on this argument. As a Native American scholar of environment and religion, I seek to understand the relationship between people and the natural world.
Native Americans view nature through their belief systems. A river or water does not only sustain life – it is sacred.
Why is water sacred to Native Americans?
In the past year, the Lakota phrase "Mní wičhóni," or "Water is life," became a new national protest anthem.
It was chanted by 5,000 marchers at the Native Nations March in Washington, D.C. this spring, and during protests last year as the anthem of the struggle to stop the building of the Dakota Access Pipeline under the Missouri River in North Dakota.
There was a reason: For long years, the Lakota, the Blackfeet and the other Native American tribes understood how to live with nature. And it was based on the knowledge of how to live within the restrictions of the limited water supply of the "Great American desert" of North America.
Water as sacred place
Native Americans learned both through observation and experiment, arguably a process quite similar to what we might call science today. They also learned from their religious ideas, passed on from generation to generation in the form of stories.
I learned from my grandparents, both members of the Blackfeet tribe in Montana, about the sacredness of water. They shared that the Blackfeet believed in three separate realms of existence – the Earth, sky and water. The Blackfeet believed that humans, or "Niitsitapi," and Earth beings, or "Ksahkomitapi," lived in one realm; sky beings, or "Spomitapi," lived in another realm; and underwater beings, or "Soyiitapi," lived in yet another. The Blackfeet viewed all three worlds as sacred because within them lived the divine.
The water world, in particular, was held in special regard. The Blackfeet believed that in addition to the divine beings, about which they learned from their stories, there were divine animals. The divine beaver, who could talk to humans, taught the Blackfeet their most important religious ceremony. The Blackfeet needed this ceremony to reaffirm their relationships with the three separate realms of reality.
The Soyiitapi, divine water beings, also instructed the Blackfeet to protect their home, the water world. The Blackfeet could not kill or eat anything living in water; they also could not disturb or pollute water.
The Blackfeet viewed water as a distinct place – a sacred place. It was the home of divine beings and divine animals who taught the Blackfeet religious rituals and moral restrictions on human behavior. It can, in fact, be compared to Mount Sinai of the Old Testament, which was viewed as "holy ground" and where God gave Moses the Ten Commandments.
Water as life
Native American tribes on the Great Plains knew something else about the relationship between themselves, the beaver and water. They learned through observation that beavers helped create an ecological oasis within a dry and arid landscape.
As Canadian anthropologist R. Grace Morgan hypothesized in her dissertation "Beaver Ecology/Beaver Mythology," the Blackfeet sanctified the beaver because they understood the natural science and ecology of beaver behavior.
Morgan believed that the Blackfeet did not harm the beaver because beavers built dams on creeks and rivers. Such dams could produce enough of a diversion to create a pond of fresh clean water that allowed an oasis of plant life to grow and wildlife to flourish.
Beaver ponds provided the Blackfeet with water for daily life. The ponds also attracted animals, which meant the Blackfeet did not have to travel long distances to hunt. The Blackfeet did not need to travel for plants used for medicine or food, either.
Beavers were part of what ecologists call a trophic cascade, or a reciprocal relationship. Beaver ponds were a win-win for all concerned in "the Great American desert" that modern ecologists and conservationists are beginning to study only now.
For the Blackfeet, Lakota and other tribes of the Great Plains, water was "life." They understood what it meant to live in a dry arid place, which they expressed through their religion and within their ecological knowledge.
Rights of rivers
Indigenous people from around the world share these beliefs about the sacredness of water.
The government of New Zealand recently recognized the ancestral connection of the Maori people to their water. This past spring, the government passed the "Te Awa Tupua Whanganui River Claims Settlement Bill," which provides "personhood" status to the Whanganui River, one of the largest rivers on the North Island of New Zealand. This river has come to be recognized as having "all the rights, powers, duties, and liabilities of a legal person" – something the Maori believed all along.
The United States does not have such laws. This new lawsuit hopes to change that and give the Colorado River "personhood" status. Indigenous people would add, a river is more than a "person" – it is also a sacred place.
(Rosalyn R. LaPier is associate professor of environmental studies at the University of Montana. This article was originally published on The Conversation. Read the original article)
TagsBlackfeet Bolivia Colombia Colorado Deep Green Resistance Ecuador India Lakota Mní wičhóni Montana Native American Native American spirituality New Zealand
Bruce Patterson says:
The world understands the personhood of water, yet cannot understand the personhood of a baby in the womb.The old word borne, is to be carried on the water. Like the Rule of Law was conceived before the flood and was borne on the water to this side of the flood, a child is conceived and borne on the water before the mother gives birth.
Givethedogabone says:
Actually I doubt very much that "the world understands the personhood of water". And no – borne does not mean carried on the water. Borne means carried; on water, on the wind, in someone's arms etc..
Your third sentence fails because it relies on the first two and they are not clearly correct (also it's not exactly rational is it?).
Look – I get that water is a vital asset; I get that it needs to be protected and managed for the benefit of all that depends upon it. I get it.
What I don't get is why some rather odd beliefs, unsupported by evidence or rational argument, mean that water becomes personified ( Important – yes, vital – yes, but personhood? – how the heck does one go from "Regarded as too valuable to be interfered with; sacrosanct" (sacred) to "The quality or condition of being an individual person" (personhood – / – "A human being regarded as an individual" (person)).
Human beings may be largely water but water is not largely human.
And yes – I guess that this is a convoluted way of trying to invoke protection via legislation that wasn't intended to be relevant. Clever, but is it sensible? What else might be accorded personhood because someone claims to have conviction unsupported by evidence or reason – air? soil? – do we have to breathe in a respectful way or plough without cutting the soil's skin?
PsiCop says:
So … people should be awarded control over things, on the grounds that they believe those things are "sacred"? The ramifications of that reasoning are, quite simply, staggering. Has anyone other than myself considered that?
Ananth Sethuraman says:
The author is trying to describe Native American concepts or institutions using English words rather than Native American words.
For example, take the word "sacred", together with its antonym "profane". These two words have force in a culture that has the concept of Satan. The same two words lose force when they are used in a non-Abrahamic culture that does have the concept of Satan.
Likewise, the words "rights" and "personhood" acquired a specific meaning because Western law discussed these words for many years in order to solve complicated legal tangles. Those legal tangles did not take place in the Native American culture, so it is likely that "rights" and "personhood" may not describe the Native American scenario properly.
Perhaps the author could use Native American words directly, rather than using "sacred", "rights" and "personhood". The meanings of the Native American words could be explained in footnotes.
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redpajama_set_name: RedPajamaCommonCrawl


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Little Canada is a city in Ramsey County, Minnesota, United States. It is an inner ring suburb of the Twin Cities. The population was 9,773 at the 2010 census. In 1844, French Canadian settler Benjamin Gervais moved north from St. Paul to claim land in order to build the first grist mill in Minnesota that was independent from the government. Today, the large lake on the east side of Little Canada bears his name (Lake Gervais). The grist mill was converted into a park, which is recognized as the birthplace of the city. Little Canada began as the township of New Canada in 1858. In the 1950s the township was threatened by the suburban sprawl of the ensuing larger communities that were formed, such as Maplewood. In 1953, the city leaders came together and established the village of Little Canada. It became a city in 1974.
From Wikipedia, the free encyclopedia. Information is reliable, but not guaranteed.
City of Little Canada
Roseville Area Public Schools
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barnes and noble east 86th street closing
1,969 people like this. But opting out of some of these cookies may have an effect on your browsing experience. Online & In Stores . Barnes and Noble New York, 2289 Broadway NY 10024 store hours, reviews, photos, phone number and map with driving directions. Barnes & Noble isn't pulling out of the Upper East Side entirely, as the company is searching for a new location in the neighborhood, a company spokesperson said. Barnes & Noble recently notified the state's Department of Labor that it plans to lay off all 64 employees at its 86th Street location in September. Barnes & Noble, which had struggled during Amazon's more than two decade-long rise, recently centered a turnaround on a strategy to localize its … While Barnes & Noble has reopened locations in other parts of the city, including in Union Square and on the Upper West Side, Daunt says the Upper East Side closing came as a surprise. In a letter sent to Barnes & Noble's 23,000+ employees on March 17, prior to closing the majority of their stores, CEO James Daunt warned that layoffs may be inevitable. "The store has served us well over the years but is now too large, and too expensive, for our needs," a Barnes & Noble spokesperson said in a statement. This Barnes & Nobles is huge. Executives in New York decided which titles to carry. While Barnes & Noble has recently suffered declining foot traffic, it still has a store at the corner of 82nd and Broadway on the Upper West Side, and in July 2009, it opened a 50,000-square-foot superstore at 86th Street and Lexington Avenue. Out of these cookies, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. The closing of a four-floor bookstore down the block from Lincoln Center could signal the beginning of the end for big box bookstores in New York. House Pushes For Trump's Quick Impeachment After Capitol Seige. Target Corp. has signed leases for new stores on the Upper East Side and the Upper West Side. Tuesday, November 5, 2019 at 7:00 PM – 8:00 PM EST. Explore current Barnes & Noble featured coupons, deals, sales, and promotions to discover great savings! WALNUT CREEK -- The Barnes and Noble bookstore on South Main Street has closed, seemingly another victim of the city's hugely successful commercial district and the soaring costs of leasing space there. To our Upper East Side community: at the request of local health officials, we are temporarily closing our 86th and Lexington location. For full functionality of this site it is necessary to enable JavaScript. Target recently signed two new leases at Upper East Side and Upper West Side locations. clock. Save this home. Shop Now. Urban Outfitters -1511 3rd Avenue. ... Barnes & Noble… Enjoy Exclusive Member Benefits. barnes & noble cafe upper east side new york; About; Blog; Businesses; Cities; Developers; Help; Careers; Cookies (Updated) ... 150 E 86th St (at Lexington Ave) New York, NY 10028 United States. A majority of the cafes are closed along with the stores. However, cafes that are open are serving on a To-Go or BN Cafe App pick-up basis only. In most of the stores, including many of the stores that are closed, B&N is offering curbside pickup for online orders. To our Upper East Side community: at the request of local health officials, we are temporarily closing our 86th and Lexington location. This category only includes cookies that ensures basic functionalities and security features of the website. The bookstore is closing its location at 150 East 86th Street after 12 years, and one of two new Targets headed for Manhattan will take its place. Leisure Living - La Habra. You also have the option to opt-out of these cookies. Barnes and Noble has been closing 15-20 bookstores all over the US for the last ten years. Friends of the Library has a few locations around the island, but Barnes and Noble will be missed by visitors and the residents that live on Maui. Shop music, movies, toys & games, too. We take the health of our booksellers and customers very seriously. Last August it was acquired by Elliot Management, after years of its sales being eroded by Amazon. Posted: (2 days ago) barnes and noble careers jobs employment application. This branch of Barnes & Noble is right near the 14 Street Union Square train station and Union Square park, putting it in a high foot traffic location. Then Barnes & Noble moved onto Lexington between 86th and 87th Streets. Littman Jewelers - Paramus. Barnes & Noble - 86th & Lexington Ave at 150 East 86th Street in New York, New York 10028: store location & hours, services, holiday hours, map, driving directions and more Shop Now. Barnes & Noble Covid-19 Updates Updated 3/31: Many of the stores are closed, and stores that remain open have reduced hours. 50% Off Select Books! FREE Curbside Pickup. We find 223 Barnes and Noble locations in New York. Community See All. 150 East 86th St, New York City +1 212-369-2180. Julie Rhinehart The Corcoran Group. Barnes & Noble once played the role of disruptor, leading to the demise of many an independent bookseller. The vast majority of them are unprofitable due to sagging print sales and the rise of e-books. We thank also the wonderful booksellers of the Upper East Side," a Barnes & Noble spokesperson said in a statement. Online & In Stores. Add reviews and photos for Barnes & Noble. Come visit our Bay Street - Emeryville location at Bay Street-Emeryville, Emeryville, CA 94608. Barnes & Noble is the nation's largest retail bookseller, and a leading retailer of content, digital media and educational products. Ulta -188 East 86th Street, Between Lex and 3rd. On top of trimming the ranks of corporate book buyers and district managers, Mr. Daunt closed some of Barnes & Noble's most iconic branches, including the East 86th Street outpost in Manhattan that housed one of New York City's most impressive art-book selections. Barnes & Noble announced earlier this week that it would be permanently closing that location and that it's looking for a smaller location nearby, according to Patch. © … See All. 2,102 people follow this. Spend $35 Get FREE Shipping. ... Barnes and Noble - New York 150 East 86th Street . Barnes & Noble Covid-19 Updates Updated 3/31: Many of the stores are closed, and stores that remain open have reduced hours. The beleaguered company remains the nation's largest bookseller. Barnes and Noble Holiday Hours. Join Now. Priority seating … Find out where to park near Barnes & Noble Booksellers 86th & Lexington and book a space. The East 86th Street Barnes & Noble, located between Third and Lexington avenues, is "too large, and too expensive" for the company's needs, a spokesman said in a statement. Home with someone with memory issues? Use the Barnes & Noble store locator to find stores and events in your area and online. The Harrisonburg Barnes & Noble is slated to close in January, according to a statement from Frank Morabito, vice president of stores for the bookstore chain. Add to wishlist Add to compare Share #445 of 17339 cafes in New York City #2571 of 8847 coffeehouses in New York City . The nation's largest retail bookseller and a leading retailer of content, digital media and educational products. Barnes and Noble store business hours may be extended during some holiday events, like Black Friday. 1.92; Barnes and Noble - New York 160 E 54th Street . 4.58; Barnes and Noble - New York 160 E 54th Street . "It is always sad to close a store but we expect to return to the Upper East Side with a new bookstore before too long, and we are in active pursuit of a new site.". The Real Deal's newsletters give you the latest scoops, fresh headlines, marketing data, and things to know within the industry. ... Barnes and Noble - New York 150 East 86th Street . Hosted by Barnes & Noble (150 East 86th Street, New York) and Susannah Cahalan. It is mandatory to procure user consent prior to running these cookies on your website. 150 East 86th Street (2,266.68 mi) New York, NY 10028. Show Map. However, it is but one of 400 stores that have shut down due to COVID-19. A Locksmith - New York. May be closed +1 212-369-2180. Gap . The East 86th Street location, situated between Third and Lexington Avenues, will replace a Barnes & Noble store. DAYTONA BEACH — A year and a half after local elementary school students persuaded the CEO of Barnes & Noble to keep his chain's Daytona Beach location, that … (Insert crying emoji here) One of the biggest Barnes and Nobles in New York City is closing its doors. Viewing 1 - 10 of 15 stores within 50 miles of Cambridge, MA STORE LOCATION. Barnes & Noble on the Upper East Side offers many wonderful events, keep up to date and get important info on the many exciting events coming up! Barnes & Noble's online bookstore for books, NOOK ebooks & magazines. 1.32; Barnes and Noble - Edgewater 465 River Road . Add to Wishlist. Closed Now. Come visit our 82nd & Broadway location at 2289 Broadway, New York, NY 10024. Add a photo. Shop music, movies, toys & games, too. Barnes and Noble has been closing 15-20 bookstores all over the US for the last ten years. Get Directions. Target signed a 20-year, 55,614-square-foot lease at Vornado Realty Trust's 150 East 86th Street, and a 15-year, 23,562-square-foot deal at the Chetrit Group's Columbus Square complex at 795 Columbus Avenue. See parking lots and garages and compare prices on the Barnes & Noble Booksellers 86th & Lexington parking map at ParkWhiz. Necessary cookies are absolutely essential for the website to function properly. In addition to author 4.6 out of 5 stars. The mainstay retailer is eighth largest in the nation and has made good on its lease payments during the pandemic while other big chains have fallen behind — among them Bed Bath & Beyond, H&M and Gap, which closed many stores during widespread lockdowns. Sephora - 144 East 86th. You can reach Orion Jones at [email protected]. And when yet another Barnes & Noble opened at 66th Street and Broadway, it was the nail in Shakespeare & Company's coffin. Not good news at all. More than a year ago. POWER iPlay® FREE Instrument Rental and Instruction to Youth K-12, Culinary Justice with Michael Twitty and Lavada Nahon, THE VERSAILLES LARGE Forest Hills one bedroom CO-OP FOR SALE, Behind The Scoreboard: Student-Athlete Sports Successes From 2020. Despite New York City reopening retail stores for curbside pickup last week, the Barnes & Noble will remain closed. See Details. The retailer's 600-plus stores were expected to follow that blueprint. Any cookies that may not be particularly necessary for the website to function and is used specifically to collect user personal data via analytics, ads, other embedded contents are termed as non-necessary cookies. To our Upper East Side community: at the request of local health officials, we are temporarily closing our 86th and Lexington location. 11,850 check-ins. 6.25; Also Viewed. Free and clear: Vornado pays off debt at 220 CPS, "Negative surprises": Vornado execs talk retail struggles on Q2 earnings call, Bank of China issues $500M to Vornado in refi of 640 Fifth Avenue, Leisure and hospitality lost 500K jobs in December, Evolving outdoor dining regulations leave restaurants in the cold, WeWork battles Chetrit Group over Midtown lease, Unibail-Rodamco-Westfield sells Long Island mall for $29.7M. Here's What You Need To Do, 5 UES Private Schools Ranked Among NYC's Best For 2021. Considered an essential business, Target remained open along with other big-box stores such as Walmart and Home Depot. This Barnes & Nobles is huge. Just as the area gets a new neighbor in the gamut of shows this September, it's about to lose another, as the Barnes & Noble on 66th Street, a neighborhood mainstay for the past 15 years, will be closing at the beginning of next year. Barnes & Noble appears in: … This website uses cookies to improve your experience while you navigate through the website. 4.6. Target signed a 20-year lease with Vornado Realty Trust for the 55,614-square-foot retail condo at the base of 150 East 86th Street, according to property records. It is a beautiful area, especially on sunny days in the Spring and Summer and well worth the visit on its own. See Details. Weekends: Weekend Barnes and Noble store business hours are from 9:00 a.m. to 11:00 p.m. Saturday and 10:00 a.m. to 10:00 p.m. on Sunday. We find 223 Barnes and Noble locations in New York. Now, Barnes & Noble, the last chain bookstore standing, is finally buckling, too. All Barnes and Noble locations in your state New York (NY). We are having some technical difficulties. By clicking Subscribe you agree to our Privacy Policy. Visit our Barnes & Noble store pages for more details and directions. Receive free shipping with your Barnes & Noble Membership. We will let you know as soon as it's possible to reopen our doors. We use cookies on our website to give you the most relevant experience by remembering your preferences and repeat visits. However, cafes that are open are serving on a To-Go or BN Cafe App pick-up basis only. "We have truly enjoyed serving our customers from this location for the past 12 years and appreciate their loyalty and support. UPPER EAST SIDE, NY — Bookstore chain Barnes & Noble closed its Upper East Side location as the company looks for a smaller space, a company spokesperson confirmed with Patch. Barnes & Noble is the world's largest bookseller, and a leading retailer of content and educational products. Please Allow Javascript and reload this page. The vast majority of them are unprofitable due to sagging print sales and the rise of e-books. Presently, the 86th Street subway stop is filled with chain stores that wouldn't be out of place at most suburban shopping malls, including Barnes & Noble, Petco, Staples and Best Buy. Barnes & Noble is an equal opportunity and affirmative action employer. 1.96; Also Viewed. Hosted by. Barnes & Noble also operates an e-bookstore and late last year launched its Nook e-book reader. "We've closed a couple of—frankly—albatross stores," Mr. Daunt said. Barnes & Noble (150 East 86th Street, New York) Bookstore in New York, New York. Buy 1, Get 1 50% Off Books for All Ages. It has four other stores in Manhattan: 82nd Street and Broadway, 555 Fifth Avenue, 33 East 17th Street and 97 Warren Street in Tribeca. Bookseller reports it will lay off 64 employees; Target to move in. Lush - 206 A East 86th . Restaurant menu $$$$ Price range per person up to $10. Barnes & Noble Location (212) 369-2180. Barnes & Noble offers books, NOOKs, textbooks, toys and games, collectibles, movies, music and more. Elliot also acquired the UK's largest bookstore chain, Waterstones, which shares a CEO with Barnes & Noble. Barnes & Noble has expanded its Buy Online Pickup In Store program to include Free Curbside Pickup. We also use third-party cookies that help us analyze and understand how you use this website. Barnes & Noble is the world's largest bookseller, and a leading retailer of content and educational products. A year ago, John Radford had little control over the book selection at the Barnes & Noble store he manages in Idaho Falls, Idaho. Find a Store ; Find an Event; Find an Event by Author; Your Location: Cambridge, MA Change Location; Stores Near My Location My B&N Stores; LIST MAP. Contact your local store for eligibility. Barnes and Noble - Rowland Heights. Add a photo + 15 photos + 14 photos + 12 photos. This is "2017 TRE Barnes & Noble East 86th Street" by lynn moore on Vimeo, the home for high quality videos and the people who love them. Come visit our 82nd & Broadway location at 2289 Broadway, New York, NY 10024. As of March 7, 2019, the company operates 627 retail stores in all 50 U.S. states. The East 86th Street Barnes & Noble, located between Third and Lexington avenues, is "too large, and too expensive" for the company's needs, a spokesman said in a … I always enjoyed browsing the Hawaiian section at Barnes and Noble. Barnes & Noble, 150 East 86th St; Barnes & Noble. Barnes & Noble - Careers. $2,100,000 Carnegie Hill 108 East 86th Street New York, NY 10028 3 Beds3 Baths Co-op. Hide Map. A sign on the door thanks patrons and suggests visiting its other East Bay stores. Barnes & Noble's online bookstore for books, NOOK ebooks & magazines. In most of the stores, including many of the stores that are closed, B&N is offering curbside pickup for online orders. Barnes & Noble has been acquired by the hedge fund Elliott Advisors for $638 million, a move that has momentarily calmed fears among publishers and … Customers and employees are required to wear masks and stay six feet away from each other. pin. Contact Barnes & Noble on Messenger. We take the health of our booksellers and customers very seriously. "Barnes and Noble was devastating," said Lenny Golay, a friend who is a … REFRESH your home for the NEW YEAR with HAVEN! Barnes and Noble New York, 150 East 86th Street NY 10028 opening times, reviews, photos, phone number and map with driving directions. New Star Cafe - New York. Barnes and Noble - 150 East 86th It is a beautiful area, especially on sunny days in the Spring and Summer and well worth the visit on its own. See parking lots and garages and compare prices on the Barnes & Noble Booksellers 86th & … Round-up of top news and topics for each of the following cities: Editorially-curated headlines and news to know for the following areas: The numbers behind office leases, retail, sales and financing. Virtual paid study starting, Coronavirus Vaccine Expands To NY Elderly, Teachers, Gunman Fires Shots Inside UES Hospital | Patch PM, Coronavirus Rates Rise | Upper East Side Week In Review, New York's New Eviction Moratorium Protections Are Not Automatic. SELLER'S REPRESENTATIVES. The store is located at 150 East 86th Street, between Lexington and Third Avenues on the Upper East Side. The big picture: The arc of retail has been bending toward consolidation for decades. Presently, the 86th Street subway stop is filled with chain stores that wouldn't be out of place at most suburban shopping malls, including Barnes & Noble, Petco, Staples and Best Buy. Many an author touches down at Barnes & Noble's citywide branches, which are fairly uniform in their size and wide selection of new books. Barnes & Noble in New York, NY -- Get driving directions to 240 E 86th St New York, NY 10028. At 23,000 square feet, the "small-format" store is about half the size of the planned East 86th Street location. Get Directions. Bookstore (212) 369-2180. crm2278@bn.com . All rights reserved © 2021 The Real Deal is a registered Trademark of Korangy Publishing Inc. Barnes & Noble is the world's largest bookseller, and a leading retailer of content and educational products. Barnes and Noble New York, 150 East 86th Street NY 10028 store hours, reviews, photos, phone number and map with driving directions. "It is always sad to close a store but we expect to return to the Upper East Side with a new bookstore before too long, and we are in active pursuit of a new site.". A majority of the cafes are closed along with the stores. Barnes and Noble Edgewater, 465 River Road NJ 7020 store hours, reviews, photos, phone number and map with driving directions. These cookies will be stored in your browser only with your consent.
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barnes and noble east 86th street closing 2021