Proposition 13

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

(i) $\exp_G$ is an isomorphism of the Lie group associated with $L(G)$ onto $G$.

(ii) Every integral subgroup of $G$ is a simply connected Lie subgroup of $G$.

Let $g = L(G)$, which is nilpotent (Proposition 12). As two simply connected Lie groups over $\mathbf{R}$ or $\mathbf{C}$ which have the same Lie algebra are isomorphic ($§ 6$, no. 3, Theorem 3 (ii)), it suffices to prove the proposition when $G$ is the group associated with $g$. Then (i) and (ii) follow from what was said before the proposition.