On real forms of complex semisimple Lie algebras

Abstract

We discuss various properties of real forms, conjugations, and automorphisms of complex semisimple Lie algebras \( \frak g \). Some of these are well known, some not so well known, and some are new. For instance we show that the group \( G^\# \) of all automorphisms of \( \frak g \), considered as a real Lie algebra, is a semidirect product of the group G of all automorphisms of \( \frak g \), considered as a complex Lie algebra, and a finite elementary abelian 2-group \( \Gamma \). We exhibit several analogies between the compact real forms on one hand and the split real forms on the other hand. We also study pairs (and triples) of commuting conjugations of \( \frak g \) with some additional properties.