Proposition 1

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

Lie Groups and Lie Algebras, chapter V, § 6. The Coxeter transformation

All Coxeter transformations are conjugate in W.

Since W permutes the chambers determined by $\mathfrak{H}$ transitively (§ 3, no. 2, Th. 1), we are reduced to proving the following: let $(C, (H_i)_{1 \leq i \leq l})$ be an ordered chamber and $\pi \in S_l$ ; then $s_{H_1} s_{H_2} \cdots s_{H_l}$ and $s_{H_{\pi(1)}} s_{H_{\pi(2)}} \cdots s_{H_{\pi(l)}}$ are conjugate in W. Taking into account § 4, no. 8, Prop. 8, this will follow immediately from the following lemma:

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

Cited by

§ 6. The Coxeter transformation
Lemma 1, § 6

Cites

§ 3, no. 2, Th. 1
§ 4, no. 8, Prop. 8