In the Wikipedia page for the Closed subgroup theorem, https://en.wikipedia.org/wiki/Closed_subgroup_theorem, there are several sufficient conditions for a subgroup $H$ of $G$ to be closed.
One of them says that if $[X,\mathfrak{h}]$ is not a subset of $\mathfrak{h}$ for all $X\in \mathfrak{g}\setminus \mathfrak{h}$ (I think this means 'self-normalising'), then $e^{\mathfrak{h}}$ is closed in $G$. The reference is to an exercise in Rossmann's 'Lie Groups – An Introduction Through Linear Groups' book.
I'm interested in finding the proof of this result. I have no clue how to begin proving it myself, and I don't have access to this textbook through my University's library, so I was wondering whether this is referenced anywhere else? Perhaps the proof is quite straightforward since it is listed as an exercise?