Proposition 1

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

Lie Groups and Lie Algebras, chapter IV, § 2. Tits Systems

Let X be a subset of S and N' a subgroup of N whose image in W is equal to W_X. Then, (G_X, B, N', X) is a Tits system.

We have $G_X = BW_XB = BN'B$ , which shows that $G_X$ is generated by $B \cup N'$ . The verification of the axioms (T1) to (T4) is now immediate.

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

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