Proposition 5

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

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

Let $P_1$ and $P_2$ be two parabolic subgroups of $G$ . Then $P_1 \cap P_2$ contains a conjugate of $T$ .

By first transforming $P_1$ and $P_2$ by an inner automorphism of $G$ , we may assume that $B \subset P_1$ . Let $g \in G$ be such that $gB_2g^{-1} \subset P_2$ . By Th. 1, there exist $n \in N$ and $b, b' \in B$ such that $g = bn b'$ . Since $T$ is normal in $N$ ,

P_2 \supset gB_2g^{-1} = bnBn^{-1}b^{-1} \supset bnTn^{-1}b^{-1} = bTb^{-1}

and

P_1 \supset B \supset bTb^{-1},

which proves the proposition.

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

Cited by

Nothing.

Cites

Th. 1