Corollary 1

Let G be a Lie group germ, G' a Lie subgroup germ of G and $\phi$ an exponential mapping of G.

(i) There exists an open neighbourhood V of 0 in $L(G')$ such that $\phi|V$ is an isomorphism of the manifold V onto an open neighbourhood of e in G'.

(ii) Let $x \in L(G)$. The following conditions are equivalent: (a) $x \in L(G')$; (b) $\phi(\lambda x) \in G'$ for $|\lambda|$ sufficiently small.

(i) is obtained by applying Proposition 8 to the canonical injection of G' into G and (ii) follows from (i).