Remark 1

Tag 01A4, permanent. Cite it as [Bourbaki, Tag 01A4].

Lie Groups and Lie Algebras, chapter IX, § 2. Maximal tori of compact Lie groups

There exists a compact subset K of L(G) such that exp $_G(K) = G$ . Indeed, if T is a maximal torus of G, there exists a compact subset $C\subset L(T)$ such that exp $_T(C) = T$ ; it suffices to take $K =\bigcup_{g\in G}$ (Ad $g$ ) $(C)$ .

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

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