Corollary 3

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

Lie Groups and Lie Algebras, chapter IV, § 1. Coxeter Groups

*The set $S$ is a minimal generating set of $W$ .

If $X \subset S$ generates $W$ , then $W = W_X$ and hence $X = S \cap W_X = S$ by Cor. 2.*

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

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Nothing.