Corollary 2

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

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

*For any subset $X$ of $S$ , we have $W_X \cap S = X$ .

This follows from Cor. 1 and the formula $S_s = \{s\}$ for $s$ in $S$ .

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

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