Further Results on Elementary Lie Algebras and Lie A-Algebras

Abstract

A finite-dimensional Lie algebra L over a field F of characteristic zero is called elementary if each of its subalgebras has trivial Frattini ideal; it is an A-algebra if every nilpotent subalgebra is abelian. This article is a continuation of the study of these algebras initiated by the authors in [10]. If we denote by 𝒜, 𝒢, ℰ, ℒ, Φ the classes of A-algebras, almost algebraic algebras, E-algebras, elementary algebras and φ-free algebras respectively, then it is shown that: It is also shown that if L is a semisimple Lie algebra all of whose minimal parabolic subalgebras are φ-free then L is an A-algebra, and hence elementary. This requires a number of quite delicate properties of parabolic subalgebras. Finally characterisations are given of E-algebras and of Lie algebras all of whose proper subalgebras are elementary.