Further results on elementary Lie algebras and Lie A-algebras

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 paper is a continuation of the study of these algebras initiated by the authors in `Elementary Lie Algebras and Lie A-algebras', J. Algebra 312 (2007), 891--901. If we denote by $\mathcal{A}$, $\mathcal{G}$, $\mathcal{E}$, $\mathcal{L}$, $\Phi$ the classes of $A$-algebras, almost algebraic algebras, $E$-algebras, elementary algebras and $\phi$-free algebras respectively, then it is shown that: \mathcal{L} \subset \Phi \subset \mathcal{G}, \mathcal{L} \subset \mathcal{A} \subset \mathcal{E} and \mathcal{G} \cap \mathcal{A} = \mathcal{L}. It is also shown that if $L$ is a semisimple Lie algebra all of whose minimal parabolic subalgebras are $\phi$-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.