On n-maximal subalgebras of Lie algebras

A chain $S_0 < S_1 < \ldots < S_n = L$ is a {\em maximal chain} if each $S_i$ is a maximal subalgebra of $S_{i+1}$. The subalgebra $S_0$ in such a series is called an {\em $n$-maximal} subalgebra. There are many interesting results concerning the question of what certain intrinsic properties of the maximal subalgebras of a Lie algebra $L$ imply about the structure of $L$ itself. Here we consider whether similar results can be obtained by imposing conditions on the $n$-maximal subalgebras of $L$, where $n>1$.