Classification of embeddings of abelian extensions of [formula omitted] into [formula omitted

Abstract

An abelian extension of the special orthogonal Lie algebra Dn is a nonsemisimple Lie algebra Dn⨭V, where V is a finite-dimensional representation of Dn, with the understanding that [V,V]=0. We determine all abelian extensions of Dn that may be embedded into the exceptional Lie algebra En+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 En+1 to Dn⨭V, and discuss which of these restrictions are or are not indecomposable.