Corollary 1

Let G be a finite-dimensional connected real or complex Lie group. For G to be nilpotent, it is necessary and sufficient that every element of Ad G be nilpotent.

If every element of Ad G is unipotent, Ad G is nilpotent (Proposition 18) and hence G, which is a central extension of Ad G, is nilpotent. If G is nilpotent, $L(G)$ is nilpotent, hence $\operatorname{ad} x$ is nilpotent for all $x \in L(G)$ and hence $\operatorname{Ad}(\exp x) = \exp \operatorname{ad} x$ is unipotent; but every element of G is of the form $\exp x$ for some $x \in L(G)$ (Proposition 14).