On the Decomposition of an Infinitesimal Group

Abstract

1. This note contains a second proof of a remarkable theorem, announced by W. Killing and first proved by Eugenio-Elia Levi, according to which any infinitesimal group G, in the sense of Lie, contains a subgroup which is simply isomorphic to the factor group G/Γ, where Γ is the greatest integrable invariant subgroup in G§. Also the theorem is sharpened in that it is proved for groups depending on real as well as on complex parameters. In the final section it is seen that the methods used in proving this theorem can be applied to the problem of solving certain equations which are stated in terms of any semi-simple infinitesimal group of linear operators.