Proposition 7

If the mapping $(m, l) \mapsto \sigma(l)m$ of $M \times L$ into $M$ is analytic, the group $S$, with the product manifold structure of $M$ and $L$, is a Lie group.

For $l, l'$ in $L$ and $m, m'$ in $M$,

whence the proposition.

If the conditions of Proposition 7 hold, the Lie group $S$ is called the (external) semi-direct product Lie group of $L$ by $M$ relative to $\sigma$.

Clearly the canonical injection of $L$ (resp. $M$) into $S$ is an isomorphism of $L$ (resp. $M$) onto a Lie subgroup of $S$ which we identify with $L$ (resp. $M$). The canonical mapping of $S$ onto $L$ is a Lie group morphism.

Conversely, let $G$ be a Lie group and $L$, $M$ two Lie subgroups such that the group $G$ is (algebraically) the semi-direct product of $L$ by $M$ (Algebra, Chapter I, § 6, no. 1). We write $\sigma(l)m = lml^{-1}$ for $l \in L$ and $m \in M$. Then $\sigma$ satisfies the conditions of Proposition 7. We can therefore form the semi-direct product Lie group $S$ of $L$ by $M$ relative to $\sigma$. The mapping $j: (m, l) \mapsto ml$ of $S$ onto $G$ is a group isomorphism and is analytic. If $j$ is a Lie group isomorphism, the Lie group $G$ is called the (internal) semi-direct product of $L$ by $M$ and $S$ and $G$ are identified. For all $g \in G$, we write $g = p(g)q(g)$, where $p(g) \in M$ and $q(g) \in L$. For the Lie group $G$ to be the semi-direct product of $L$ by $M$, it is necessary and sufficient that one of the mappings $p: G \to M$ and $q: G \to L$ be analytic, in which case both are analytic; or alternatively, it is necessary and sufficient that $T_e(G)$ be the topological direct sum of $T_e(M)$ and $T_e(L)$ (for, if this condition holds, $j$ is étale at $e_S$).