Proposition 7

*Let $G$ be a Lie group and $a$ an element of $L(G)$. The mapping $\lambda \mapsto \exp(\lambda a)$ of $K$ into $G$ is the unique morphism $\phi$ of the Lie group $K$ into $G$ such that $(T_0 \phi)1 = a$.

The mappings $(\lambda, \lambda') \mapsto \exp(\lambda a) \exp(\lambda' a)$ and $(\lambda, \lambda') \mapsto \exp(\lambda + \lambda') a$ of $K \times K$ into $G$ are analytic and coincide on a neighbourhood of $(0, 0)$. As $K \times K$ is connected, these mappings are equal. Hence $\phi : \lambda \mapsto \exp(\lambda a)$ is a Lie group morphism of $K$ into $G$. The tangent mapping at 0 to $\lambda \mapsto \lambda a$ is the mapping $\lambda \mapsto \lambda a$; and $T_e(\exp) = \mathrm{Id}_{L(G)}$; hence $(T_0 \phi)1 = a$. The uniqueness assertion of the proposition follows from Theorem 1.*