Corollary 1

Tag 0ACM, permanent. Cite it as [Bourbaki, Tag 0ACM].

Let $G$ and $H$ be two Lie group germs. If $L(G)$ and $L(H)$ are isomorphic, $G$ and $H$ are locally isomorphic.

This follows from Theorem 1 and § 1, no. 10, Proposition 21.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by