Corollary 1

For a Lie algebra g to be nilpotent, it is necessary and sufficient that, for all $x \in g$, $\mathrm{ad}\ x$ be nilpotent.

The condition is necessary (Proposition 1). Suppose that its sufficiency has been proved for Lie algebras of dimension $< n$ ($n \neq 0$). Let $g$ be an $n$-dimensional Lie algebra such that, for all $x \in g$, $\mathrm{ad}\ x$ is nilpotent. Theorem 1, applied to the set of $\mathrm{ad}\ x\ (x \in g)$, proves that the centre $c$ of $g$ is non-zero. Then $g$ is a central extension of the Lie algebra $g/c$, which is nilpotent by our induction hypothesis. The proof is completed by applying Proposition 2.