Corollary 1

Let G be a finite-dimensional simply connected Lie group. Let m, h be Lie subalgebras of L(G) such that L(G) is the semi-direct product of m by h. Let M, H be the corresponding integral subalgebras of G. Then M and H are simply connected Lie subgroups of G and, as a Lie group, G is the semi-direct product of M by H.

By Proposition 14, H is a normal Lie subgroup of G and the Lie group G/H is simply connected. Let $\pi$ be the canonical morphism of G onto G/H. There exists a morphism $\theta$ of G/H into M such that $L(\theta)$ is the canonical isomorphism of $L(G)/L(H) = L(G)/h$ onto $L(M) = m$. Then

and hence $\pi \circ \theta = \mathrm{Id}_{G/H}$. By no. 1, Corollary 1 to Proposition 1, $\theta(G/H) = M$. By Proposition 8 of § 1, no. 4, M is a Lie subgroup of G and the Lie group G is the semi-direct product of M by H.