This is Exercise 7.34 in Lie's book: Suppose $N$ and $H$ are Lie groups, and $\theta$ is a smooth action of $H$ on $N$ by automorphisms. Let $G=N\rtimes_\theta H$. Then I am trying to show that $\tilde N=N\times\{e\}$ and $\tilde H=\{e\}\times H$ are closed Lie subgroups of $G$ isomorphic to $N$ and $H$, respectively.
Showing this for $\tilde N$ is easy enough since $\tilde N$ is the preimage of $e$ under the projection map $(n,h)\mapsto h$. I've been having some trouble finding a solution for $\tilde H$. I tried coming up with a constant rank map which has $\tilde H$ as a level set, but I haven't been able to find one yet. I think I might be missing something obvious here.