Remark 3

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

Lie Groups and Lie Algebras, chapter V, § 3. Groups of displacements generated by reflections

Let C be a chamber of E and S the set of reflections with respect to the walls of C. Let $w \in W$ and let $(s_1, \ldots, s_q)$ be a reduced decomposition of w with respect to S. If $x \in \overline{C}$ is fixed by w, then $s_j(x) = x$ for all j: this follows from Prop. 1 above and Cor. 1 of Prop. 7 of Chap. IV, § 1.

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

Cited by

Nothing.

Cites

Cor. 1 of Prop. 7 of Chap. IV, § 1
Prop. 1