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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 a previous paper. 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} \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.

In this paper we study the class ${\cal F}$ of Lie algebras having a flag of subalgebras, and the class ${\cal Ch}_{lm}$ of Lie algebras having a maximal chain of lower modular subalgebras. We show that ${\cal F} \subseteq {\cal Ch}_{lm}$ and that both are extensible formations that are subalgebra closed. We derive a number of properties relating to these two classes, including a classification of the algebras in each class over a field of characteristic zero.

The main purpose of this paper is to study Lie algebras L such that if a subalgebra U of L has a maximal subalgebra of dimension one then every maximal subalgebra of U has dimension one. Such an L is called lm(0)-algebra. This class of Lie algebras emerges when it is imposed on the lattice of subalgebras of a Lie algebra the condition that every atom is lower modular. We see that the effect of that condition is highly sensitive to the ground field F. If F is algebraically closed, then every Lie algebra is lm(0). By contrast, for every algebraically non-closed field there exist simple Lie algebras which are not lm(0). For the real field, the semisimple lm(0)-algebras are just the Lie algebras whose Killing form is negative-definite. Also, we study when the simple Lie algebras having a maximal subalgebra of codimension one are lm(0), provided that the characteristic of F is different from 2. Moreover, lm(0)-algebras lead us to consider certain other classes of Lie algebras and the largest ideal of an arbitrary Lie algebra L on which the action of every element of L is split, which might have some interest by themselves.