Corollary 6
Tag 01AA, permanent. Cite it as [Bourbaki, Tag 01AA].
Lie Groups and Lie Algebras, chapter IX, § 2. Maximal tori of compact Lie groups
The maximal tori of G are their own centralizers, and are the fixers of their Lie algebras.
Let T be a maximal torus of G and C its centralizer; since L(T) is a Cartan subalgebra of L(G), we have L(T) = L(C), hence C = T since C is connected (Cor. 5).
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.