Proposition 48

Let G be a Lie group and H and E Lie subgroups of G. Suppose that G is, as a Lie group, the semi-direct product of H by E. Let $\varphi$ be the linear representation $g \mapsto (\mathrm{Ad}\, g) \mid L(E)$ of the Lie group G on $L(E)$ (cf. Proposition 47) and let $\sigma$ be the restriction of $\varphi$ to H. Then: (i) $L(G)$ is the topological direct sum of $L(H)$ and $L(E)$; (ii) $L(H)$ is a subalgebra of $L(G)$ and $L(E)$ is an ideal of $L(G)$; (iii) $L(\sigma)$ is a linear representation of $L(H)$ on the Lie algebra of derivations of $L(E)$; (iv) $L(G)$ is the semi-direct product of $L(H)$ by $L(E)$ defined by $L(\sigma)$ (Chapter I, § 1, no. 8).

(i) is obvious and (ii) follows from Proposition 47. $L(\sigma) = L(\varphi) \mid L(H)$. Now by Propositions 40 (no. 11) and 44 (no. 12), $L(\varphi)(t)$ is, for all $t \in L(G)$, the restriction of $\mathrm{ad}_{L(G)} t$ to $L(E)$. This proves (iii). Using (i) and (ii), this also proves (iv).