Proposition 8

Let $G$ and $H$ be Lie groups, $p: G \to H$ and $s: H \to G$ Lie group morphisms such that $p \circ s = \mathrm{id}_H$ and $N = \mathrm{Ker}\ p$. Then $N$ is a Lie subgroup of $G$, s is an isomorphism of $H$ onto a Lie subgroup of $G$ and the Lie group $G$ is the internal semi-direct product of $s(H)$ by $N$.

$T_e(p) \circ T_e(s) = \mathrm{id}_{T_{e(H)}}$ and hence $p$ (resp. $s$) is a submersion (resp. an immersion). By Differentiable and Analytic Manifolds, R, 5.10.5, N is a Lie subgroup of G. On the other hand, $s$ is a homeomorphism of H onto $s(H)$ and hence $s$ is an isomorphism of H onto a Lie subgroup of G (Differentiable and Analytic Manifolds, R, 5.8.3). Finally, for all $g \in G$, $g = (s \circ p)(g) \cdot n$ for some $n \in N$; as $s \circ p$ is analytic, the Lie group G is the semi-direct product of $s(H)$ by N.