Exercise 6, § 8

Let $G$ be a group of class $C^r$ (§ 5, Exercise 1) with $r \geqslant 2$. There exists on $G$ one and only one real Lie group structure $S$ such that the underlying manifold structure of class $C^r$ of $S$ is the given structure. (The uniqueness of $S$ follows from Corollary 1 to Theorem 1. Let $L(G)$ be the normable Lie algebra associated with $G$ in Exercise 2 of § 5. There exist a real Lie group germ $G'$ and an isomorphism $h$ of $L(G')$ onto $L(G)$ (§ 4, Theorem 3). Verify as in § 4, no. 1 that there exist a symmetric open neighbourhood $G''$ of $e_G$ in $G'$ and a mapping $\phi$ of class $C^r$ of $G''$ into $G$ such that $T_e(\phi) = h$ and $\phi(g_1g_2) = \phi(g_1)\phi(g_2)$ for $g_1, g_2$ in $G''$. By shrinking $G''$, it can be assumed that $V = \phi(G'')$ is open in $G$ and that $\phi$ is an isomorphism of class $C^r$ of the manifold $G''$ onto the manifold $V$. There therefore exists on $V$ a real Lie group germ structure such that the underlying manifold structure of class $C^r$ is the given structure. For all $g \in G$, Int $g$ defines as analytic mapping of $V \cap (g^{-1}Vg)$ onto $(gVg^{-1}) \cap V$ (Theorem 1). By Proposition 18 of § 1, there exists on $G$ a real Lie group structure $S$ inducing the same analytic structure as $V$ on an open neighbourhood of $e$. By translation, the underlying manifold structure of class $C^r$ of $S$ on $G$ is the given structure.)