Proposition 2

Let G be a real or complex Lie group and $\alpha$ *an infinitesimal automorphism of G. There exists a law of analytic operation $(\lambda, g) \mapsto \phi_\lambda(g)$ of K on G with the following properties: (1) if D is the associated law of infinitesimal operation, then $D(1) = \alpha$; (2) for all $\lambda \in K$, $\phi_\lambda \in \mathrm{Aut}\, G$. (a) For all $\mu > 0$, let $K_\mu$ be the open ball of centre 0 and radius $\mu$ in K. For all $g \in G$, let $\mathscr{F}_g$ be the set of analytic integral curves $f$ of $\alpha$ defined in a ball $K_\mu$ and such that $f(0) = g$. By Differentiable and Analytic Manifolds, R, 9.1.3 and 9.1.5, $\mathscr{F}_g$ is non-empty and two elements of $\mathscr{F}_g$ coincide on the intersection of their domains of definition; let $\mu(g)$ be the least upper bound of the numbers $\mu$ such that there exists an element of $\mathscr{F}_g$ defined in $K_\mu$; there exists a unique element of $\mathscr{F}_g$ defined in $K_{\mu(g)}$; we denote it by $f_g$. (b) Let $g_1, g_2$ be in G, $f_1 \in \mathscr{F}_{g_1}, f_2 \in \mathscr{F}_{g_2}$ with $f_1$ and $f_2$ defined on the same ball $K_\mu$. Then $f_1 f_2 : K_\mu \to G$ is analytic and $(f_1 f_2)(0) = g_1 g_2$. On the other hand, for all $\lambda \in K_\mu$,

and hence $f_1 f_2 \in \mathscr{F}_{g_1 g_2}$. This proves that $\mu(g_1 g_2) \geq \inf_{g \in V} (\mu(g_1), \mu(g_2))$. (c) By Differentiable and Analytic Manifolds, R, 9.1.4 and 9.1.5, there exists a neighbourhood V of e in G such that $\sigma = \inf_{g \in V} \mu(g) > 0$. Let $h \in G$ and C be its connected component. For all $h' \in C$, $\mu(h') \geq \inf(\sigma, \mu(h)) > 0$ by (b). On the other hand, the functions $f_{h'}$, where $h' \in C$, take their values in C. By Differentiable and Analytic Manifolds, R, 9.1.4 and 9.1.5, $\mu = +\infty$ in C and finally $\mu = +\infty$ in G. Then let $f_g(\lambda) = \phi_\lambda(g)$ for all $g \in G$ and all $\lambda \in K$. By Differentiable and Analytic Manifolds, R, 9.1.4 and 9.1.5, the mapping $(\lambda, g) \mapsto \phi_\lambda(g)$ is a law of analytic operation of K on G. Clearly, if D is the associated law of infinitesimal operation, $D(1) = \alpha$. By (b),

for all $\lambda \in K, g_1 \in G, g_1 \in G$.