Lemma 3

Let $G$ and $G'$ be Lie groups and $\phi$ a homomorphism of $G$ into $\mathrm{Aut}(G')$. Let $f(g, g') = (\phi(g))(g')$ for $g \in G, g' \in G'$. Consider the following conditions: (i) $f$ is analytic; (ii) $f$ is analytic in a neighbourhood of $(e_G, e_{G'})$; (iii) for all $g' \in G'$, the mapping $g \mapsto f(g, g')$ is analytic. Then (i) $\Leftrightarrow$ ((ii) and (iii)). If $G$ is connected, (i) $\Leftrightarrow$ (ii). Clearly (i) implies (ii) and (iii). Let $g_0 \in G, g_0' \in G'$. For all $g \in G, g' \in G'$,

This proves the implication ((ii) and (iii)) $\Rightarrow$ (i). Finally, if $G'$ is connected, $G'$ is generated by every neighbourhood of $e_{G'}$ and hence (ii) $\Rightarrow$ (iii).