Proposition 11

Let $G$ be a connected commutative Lie group. (i) exp is an étale morphism of the additive Lie group $L(G)$ onto $G$.

(ii) If K = \mathbf{R} and G is finite-dimensional, G is isomorphic to a Lie group of the form $\mathbf{R}^p \times \mathbf{T}^q$ ($p, q$ integers $\geqslant 0$).

By the Hausdorff formula, $(\exp x)(\exp y) = \exp(x + y)$ for $x, y$ sufficiently close to 0 and hence for all $x, y$ in $L(G)$ by analytic continuation. Hence $\exp$ is a group homomorphism and is étale since

Hence (i). Assertion (ii) follows from (i) and General Topology, Chapter VII, § 1, Proposition 9.