Semi-simple real subal gebras of non-compact semi-simple real lie algebras V

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 when L and/or L is exceptional. Matrix representations for all the exceptional Lie algebras are calculated. Detailed procedures are given, which, together with those given in previous papers, allow the construction of all embeddings of L in L , when their complex extensions are A 1, B 1, C 1, D 1, E 6, E 7, E 8, F 4, G 2 or a direct sum of any two of there. The procedures are illustrated by examples, including all real semi-simple Lie subalgebras of real forms of G 2 and sub-algebras of real forms of F 4 whose complex extensions are B 4 or A 1 (representation (16) + (9)). Because of its physical significance, all embeddings of SL (2, C) in real forms of F 4 and E 6 are given. Many of these are new results.