Corollary 1

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

Lie Groups and Lie Algebras, chapter VI, § 1. Root systems

Let $\beta \in B(C)$ . The reflection $s_\beta$ permutes the positive roots not proportional to $\beta$ .

We reduce immediately to the case in which R is reduced. In that case, our assertion follows from (ii) and the fact that $T_{s_\beta} = s_\beta$ .

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

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