Abstract
The main purpose of this paper is to study the finite-dimensional solvable Lie algebras described in its title, which we call minimal non-{mathcal N}. To facilitate this we investigate solvable Lie algebras of nilpotent length k, and of nilpotent length ≤k, and 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 ≤3.
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