  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 $n>1$.
