In Theorem 7.35, if we remove the closedness condition for $H$, we also can prove the map $(n,h)\mapsto nh$ is a Lie group isomorphism between $N\rtimes_{\theta}H$ and $G$, then $H$ is a closed Lie subgroup of $G$.
What I said is correct?
2026-04-03 06:05:08.1775196308
Can the closedness condition for $H$ be removed?
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You're right. This was an oversight on my part. Thanks for pointing it out. I've added a correction to my online list.