Proposition 1

Let G and M be Lie groups, H an integral subgroup of G and $\phi$ a morphism of M into G such that $L(\phi)(L(M)) \subset L(H)$. Suppose that M is connected. Then $\phi(M) \subset H$ and $\phi$, considered as a mapping of M into H, is a Lie group morphism.

In the notation of Remark 1, $\phi$ is a morphism of M into Y (Differentiable and Analytic Manifolds, R, 9.3.2) and hence $\phi(M) \subset H$ since M is connected.