Theorem 2
Tag 0ACR, permanent. Cite it as [Bourbaki, Tag 0ACR].
Lie Groups and Lie Algebras, chapter III, § 4. Passage from Lie algebras to Lie groups
Let L be a complete normable Lie algebra. There exists a Lie group germ G such that L(G) is isomorphic to L. Two such Lie group germs are locally isomorphic.
The first assertion follows from Lemmas 2 and 3. The second assertion follows from Corollary 1 to Theorem 1 of no. 1.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.