Semi-simple real subalgebras of non-compact semi-simple real lie algebras. III

Abstract

The investigation of the problem of embedding a semi-simple real Lie algebra L ' in a non-compact semi-simple real Lie algebra L is extended to the case in which at least one of the real Lie algebras has D l as its complex extension and is obtained from the compact form SO(2 l) by an outer automorphism. Detailed procedures are given, which, together with those given previously, allow the construction of all embeddings of L ' in L when their corresponding complex extensions are any of the simple classical Lie algebras. Embeddings involving real forms of D 4, which have exceptional properties, are also considered in detail. The procedures are illustrated by considering examples corresponding to the complex Lie algebra embeddings A l⋐D l+1 , l⩾2, D l⋐B l , l⩾3, and B l⋐D l+1 , l⩾2, giving some new results.