Proposition 14

Let $G$ be a finite-dimensional connected Lie group over $\mathbf{R}$ or $\mathbf{C}$.

(i) If $G$ is nilpotent, $\exp_G$ is étale and surjective.

(ii) If $K = \mathbf{C}$ and $\exp_G$ is étale, then $G$ is nilpotent.

Let $G'$ be the universal covering space of $G$. Let $\phi$ be the canonical morphism of $G'$ onto $G$. Then $\exp_G = \phi \circ \exp_{G'}$ ($§ 6$, no. 4, Proposition 10) and hence (i) follows from Proposition 13 (i).

If $K = \mathbf{C}$ and $\exp$ is étale, then, for all $x \in L(G)$, and $x$ has no eigenvalue belonging to $2i\pi(\mathbf{Z} - \{0\})$ ($§ 6$, no. 4, Corollary to Proposition 12). Applying this to $\lambda x$, where $\lambda$ varies through $\mathbf{C}$, it follows that all the eigenvalues of $\mathrm{ad}\, x$ are zero and hence that $\mathrm{ad}\, x$ is nilpotent. Therefore $L(G)$ is nilpotent (Chapter I, $§ 4$, Corollary 1 to Theorem 1) and hence $G$ is nilpotent (Proposition 12).