Remark 4

Tag 0AV0, permanent. Cite it as [Bourbaki, Tag 0AV0].

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

Assertion (vi) of Th. 2 shows that the map $C \mapsto B(C)$ is a bijection from the set of chambers to the set of bases. Consequently, $W(R)$ acts simply-transitively on the set of bases.

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

Cited by

Remark 3, § 4
Proposition 19, § 4

Cites

Th. 2