Let $\mathfrak g$ be a Lie algebra and let $\mathfrak h\subset\mathfrak j$ be two subalgebras of $\mathfrak g$. Now let $G$ be a Lie group with Lie algebra $\mathfrak g$ and let $H$ be a Lie (not necessary connected) subgroup of $G$ with Lie algebra $\mathfrak h$ and let $J$ be the connected Lie subgroup of $G$ corresponding to $\mathfrak j$.
Is it always true that $HJ:=\{hj;\ h\in H, j\in J\}$ is a Lie subgroup of $G$?
Let $G=SO(3)$, ${\mathfrak h}=0$, ${\mathfrak j}\cong {\mathbb R}$. Thus, $J\cong SO(2)$. As for $H$, take a randomly chosen (i.e. not normalizing $J$) finite subgroup of order 2. I leave you to check that $HJ$ is not a subgroup.