[Math] Normal subgroup and Lie algebra

Question

I have an exercise of Lie group as follows: "Let $G,H$ be closed connected subgroup of $GL_n(\mathbb{R})$, and $H$ be subgoup of $G$. Suppose that $\operatorname{Lie}(H)$ is an ideal of $\operatorname{Lie}(G)$. Prove that $H$ is a normal subgroup of $G$."
I get stuck to solve this problem. Also I have no idea to use the connectedness of $G$ and $H$. Some one can help me? Thanks a lot!

Answer 1

This is essentially an application of the Lie subalgebra-subgroup correspondence:

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. Suppose that $\mathfrak{h}$ is a Lie subalgebra of $\mathfrak{g}$. Then there is a unique connected immersed Lie subgroup $H\subseteq G$ whose Lie algebra corresponds to $\mathfrak{h}$.

You are given $H$ a closed connected Lie subgroup of the Lie group $G$. Choose $g\in G$, and let $H' = gHg^{-1}$. Then $H'$ is a Lie group with corresponding Lie algebra $Lie(H')\subseteq Lie(G)$. However, the assumption that $Lie(H)$ is an ideal of $Lie(G)$ says exactly that $Lie(H') = Lie(H)$. You then have that $H$ and $H'$ are two connected Lie subgroups of $G$ with the same Lie algebra. By the uniqueness in the above theorem, it follows that $H = H'$, and hence that $H$ is normal.