Commutator groups and algebras

Abstract

Let Hand K be connected, Lie subgroups of a Lie group C. The group [H , K] , generated by all commutators hkh1k l(h€H, k€K) is arcwise connected. Therefore, by a theorem of Yamabe, LH, K] is a Lie subgroup. If ~, Sl' denote the Lie algebras of Hand K , respectively , then the Lie algebra of [H , K] is the s mallest algebra containing [~ , S\\ ], which is invariant underad~ andadSl'. An immediate consequ ence is that if Hand K are complex Lie subgroups, then [H , K] is also complex.