Exercise 7, § 8

Let $G$ be a real Lie group and $H$ a closed subgroup of $G$. (a) Let $\mathfrak{h}$ be the set of $x \in L(G)$ such that $\exp(tx) \in H$ for all $t \in \mathbf{R}$. Then $\mathfrak{h}$ is a Lie subalgebra of $L(G)$. (Use Proposition 8 of § 6.)

† For more details, cf. M. Goto, On an arcwise connected subgroup of a Lie group, Proc. Amer. Math. Soc., 20 (1969), pp. 157–162.

(b) Suppose that H is locally compact. Show that H is a Lie subgroup of G. (Show first that $\mathfrak{h}$ is finite-dimensional by proving the existence in $\mathfrak{h}$ of a precompact neighbourhood of 0. Then imitate the proof of Theorem 2, choosing $V_2$ such that $(\exp V_2) \cap H$ is relatively compact.)