Proposition 21

Let $G, H$ be two Lie group germs and $\phi$ a morphism of $G$ into $H$. The following conditions are equivalent:

(i) $\phi$ is étale at $e$; (ii) there exist open Lie subgroup germs $G', H'$ of $G, H$ such that $\phi|G'$ is an isomorphism of $G'$ onto $H'$.

The implication (ii) $\Rightarrow$ (i) is obvious. Suppose that $\phi$ is étale at $e$. There exists an open Lie subgroup germ $G_1$ of $G$ such that $\phi(G_1)$ is open in $H$ and $\phi|G_1$ is an isomorphism of the manifold $G_1$ onto the manifold $\phi(G_1)$. Then there exists an open Lie subgroup germ $G'$ of $G_1$ such that the product in $G$ of two elements of $G'$ is always defined and belongs to $G_1$. If $g, g'$ are elements of $G'$ such that $gg' \in G'$, then $\phi(g)\phi(g') = \phi(gg') \in \phi(G')$; if $g, g'$ are elements of $G'$ such that $gg' \in G_1 - G'$, then

Hence $\phi|G'$ is an isomorphism of the Lie group germ $G'$ onto the open Lie subgroup germ $\phi(G')$ of $H$.

If the conditions of Proposition 21 hold, $G$ and $H$ are called locally isomorphic.