  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.
