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.
Categories: math.RA math.RT
