Lie algebras with nilpotent length greater than that of each of their subalgebras

Abstract

The main purpose of this paper is to study the finite-dimensional solvable Lie algebras described in its title, which we call {\em minimal non-$\mathcal{N}$}. To facilitate this we investigate solvable Lie algebras of nilpotent length $k$, and of nilpotent length $\le k$, and {\em extreme} Lie algebras, which have the property that their nilpotent length is equal to the number of conjugacy classes of maximal subalgebras. We characterise the minimal non-$\mathcal{N}$ Lie algebras in which every nilpotent subalgebra is abelian, and those of solvability index $\le 3$.