From P210 of "Linear algebraic groups and finite groups of Lie type" by Malle and Testerman.
Theorem 24.10 Let $G$ be connected reductive with Steinberg endomorphism $F : G → G$, $T ≤ B$ an $F$ -stable maximal torus in an $F$ -stable Borel subgroup of $G$, $N := N_G(T)$. Then $B^F$ , $N^F$ is a BN-pair in $G^F$ with Weyl group $W^F$ .
Corollary 24.11 In the situation of Theorem 24.10, let $U = R_u(B)$. Then $U^F$ is a Sylow $p$-subgroup of $G^F$, with normalizer $N_{G^F} (U^F) = B^F$.
In the proof for this corollary, it states $N_{G^F}(U^F)$ is a parabolic subgroup of $G^F$. By(BN5), it does not contain any simple reflection, so it equals $B^F$.
By the way, BN5 is: If $\dot{s}$ is in N, then $\dot{s}$B$\dot{s}$ $\neq$ B.
Why it has no simple reflection? Thank you!
