Bourbaki

Corollary 2

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

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

Let w∈Ww \in W and let TwT_w be the subset of WW associated to ww by the procedure of Lemma 2 of § 1, no. 4. If t∈Twt \in T_w, then

C(t)⊂C(w).C(w−1). C(t) \subset C(w).C(w^{-1}).

If t∈Twt \in T_w, there exist by definition elements w′,w′′∈Ww', w'' \in W and s∈Ss \in S such that

w=w′sw′′,lS(w)=lS(w′)+lS(w′′)+1andt=w′sw′−1. w = w' s w'', \quad l_S(w) = l_S(w') + l_S(w'') + 1 \quad \text{and} \quad t = w' s {w'}^{-1}.

By Cor. 1,

C(w).C(w−1)=C(w′).C(s).C(w′′).C(w′′−1).C(s).C(w′−1). C(w).C(w^{-1}) = C(w').C(s).C(w'').C({w''}^{-1}).C(s).C({w'}^{-1}).

Hence,

C(w).C(w−1)⊃C(w′).C(s).C(s).C(w′−1). C(w).C(w^{-1}) \supset C(w').C(s).C(s).C({w'}^{-1}).

By (4), C(s)⊂C(s).C(s)C(s) \subset C(s).C(s). Hence,

C(w).C(w−1)⊃C(w′).C(s).C(w′−1)⊃C(t). C(w).C(w^{-1}) \supset C(w').C(s).C({w'}^{-1}) \supset C(t).

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

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Transcribed from the printed volume. Not translated.