Remark 3

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

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

Since the $\alpha \tilde{}$ generate $V^*$ , the group $W(R)$ , considered as a subgroup of $\mathbf{GL}(V_{\mathbf{R}})$ , is essential (Chap. V, § 3, no. 7). Moreover, the Cor. of Th. 1 of Chap. V, § 3, no. 2 shows that the only reflections belonging to $W(R)$ are the $s_{\alpha}$ .

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

Cited by

Theorem 2, § 1

Cites

Chap. V, § 3, no. 7
Cor. of Th. 1 of Chap. V, § 3, no. 2