When the identity component of an abstract subgroup is a Lie subgroup

Question

If $G$ is a Lie group and $H$ is an abstract subgroup of $G$ such that the identity component $H^0$ is a Lie subgroup of $G$. Is $H$ a Lie subgroup of $G$?

Answer 1

In general, no. Take, for instance, $G=(\mathbb R^2,+)$ and $H=\mathbb R\times\mathbb Q$. Then $H^0=\mathbb R\times\{0\}$, which is a $1$-dimensional Lie group, but $H$ is not a Lie subgroup of $G$.