Exercise 22, § 6

Let $G$ be a real Lie group, $H$ a normal 1-dimensional integral subgroup and $A$ a closed subgroup of $G$ such that $AH$ is not closed in $G$. Then $H$ is central in the identity component of $\overline{AH}$. (Reduce it to the case where $G = \overline{AH}$ and $G$ is connected. Then the set $B$ of elements of $A$ which commute with $H$ is a normal subgroup of $G$. Passing to the quotient by $B$, reduce it to the case $B = \{e\}$. As every commutator in $G$ commutes with $H$, $A$ is then commutative. By assuming that $H$ is closed and using Exercise 21 (c), show that $AH$ would be closed, whence a contradiction. Hence $H$ is not closed and Exercise 21 (a) can be used.)