Classification of embeddings of abelian extensions of $D_n$ into $E_{n+1}$

An abelian extension of the special orthogonal Lie algebra $D_n$ is a nonsemisimple Lie algebra $D_n \inplus V$, where $V$ is a finite-dimensional representation of $D_n$, with the understanding that $[V,V]=0$. 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. We then classify these embeddings, up to inner automorphism. As an application, we also consider the restrictions of irreducible representations of $E_{n+1}$ to $D_n \inplus V$, and discuss which of these restrictions are or are not indecomposable.

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