Proposition 14

Let $G$ be a Lie group and $H$ an integral subgroup whose topology admits a countable base. The following conditions are equivalent: (i) $H$ is normal in $G$; (ii) $L(H)$ is invariant under $\mathrm{Ad}(G)$. If further $G$ is connected, these conditions are equivalent to the following: (iii) $L(H)$ is an ideal of $L(G)$. If further $G$ is simply connected and $L(H)$ is of finite codimension in $L(G)$, these conditions imply that $H$ is a Lie subgroup of $G$ and that $G/H$ is simply connected.

The equivalence (i) $\Leftrightarrow$ (ii) follows from Lemma 3. If further $G$ is connected, condition (ii) is equivalent to saying that $L(H)$ is stable under $\mathrm{ad}\, L(G)$ (no. 5, Corollary 1 to Proposition 13 and § 3, no. 12, Proposition 44).

Suppose that $G$ is simply connected and that $L(H)$ is an ideal of finite codimension in $L(G)$. By Theorem 3 of no. 3, there exists a simply connected Lie group $G'$ such that $L(G')$ is isomorphic to $L(G)/L(H)$. There exists a continuous morphism $f$ of $L(G)$ onto $L(G')$ with kernel $L(H)$. By Theorem 1 of no. 1, there exists a morphism $\phi$ of $G$ into $G'$ such that $L(\phi) = f$. This morphism is a submersion and hence its kernel $N$ is a Lie subgroup of $G$ such that $L(N) = \mathrm{Ker}\, f = L(H)$. Hence $H$ is the identity component of $N$ and is therefore a Lie subgroup of G. Let $\psi$ be the morphism of $G/H$ into $G'$ derived from $\phi$ when passing to the quotient. This morphism is étale; since G is simply connected, $\psi$ is an isomorphism of $G/H$ onto $G'$.