How to prove that the adjoint group is a Lie subgroup of $Gl(\mathfrak{g})$ Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. Let $Ad: G \rightarrow GL(\mathfrak{g})$ be the Adjoint representation. I want to pro...--prophetes.ai

How to prove that the adjoint group is a Lie subgroup of $Gl(\mathfrak{g})$ Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. Let $Ad: G \rightarrow GL(\mathfrak{g})$ be the Adjoint representation. I want to prove that $Ad(G)$ is a Lie subgroup of $GL(\mathfrak{g})$. Here is what I tried: From Cartan's Theorem, it suffices to show that $Ag(G)$ is closed in $GL(\mathfrak{g})$. So take $x \in Ad(G)$ and $(Ad(g_n))_n$ a sequence in $Ad(G)$ that converges to $x$, with $g_n \in G$. I can't continue from here. It may be possible that some additional other hypothesis are required. Thanks!

User proposition 7.1 in San Martin book. <