Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. Let $H$ be a Lie subgroup of $G$ and its Lie algebra $\mathfrak{h}$ is naturally a subalgebra of $\mathfrak{g}$. Let $G^\circ$ denote the identity component of $G$, $H^\circ$ denote the identity component of $H$. Assume now $\mathfrak{h}$ is an ideal of $\mathfrak{g}$, then we know $H^\circ$ is a normal subgroup of $G^\circ$. Is it true that $H^\circ$ is also a normal subgroup of $G$?
I think it is highly possible that this is not true, but I cannot find a counter example since one need to find a suitable unconnected Lie group here.