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An abelian extension of the special orthogonal Lie algebra $D_n$ is anonsemisimple Lie algebra $D_n \inplus V$, where $V$ is a finite-dimensionalrepresentation of $D_n$, with the understanding that $[V,V]=0$. We determineall abelian extensions of $D_n$ that may be embedded into the exceptional Liealgebra $E_{n+1}$, $n=5, 6$, and 7. We then classify these embeddings, up toinner automorphism. As an application, we also consider the restrictions ofirreducible representations of $E_{n+1}$ to $D_n \inplus V$, and discuss whichof these restrictions are or are not indecomposable.

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