We note $G$ a connected algebraic group, $T$ a maximal torus in $G$, $B$ a Borel subgroup containing $T$. We put also, $N=N\_{G}(T)$ the normalizer of $T$ in $G$. We know that $(B,N)$ is a BN-pair of $G$. How we prove that every parabolic subgroup of $G$ containing $B$ is of the form $P=BW\_{T}B$, where $W$ is the Weyl group of the BN-pair $(B,N)$, it's the same as a Weyl group of $G$, and $W\_{T}=<T>$ a standard subgroup of $W$ ($W$ is a Coxeter group with set of generators $S$, and $T$ is a subset of $S$ and finally $W\_{T}$ is a subgroup generated by $T$). 



Parabolic subgroups and BN-pairs
Combine Theorem 6.43 (1) pg. 315 in Abramenko-Brown "Buildings" together with Proposition 6.36 (6) on pg. 310.


These are two steps:
As you mention, $BW\_T B$ is a subgroup (Proposition 6.36).


$BW\_TB$ contains $B$, any subgroup containing the Borel subgroup is parabolic, and any parabolic is conjugate to a subgroup containing a specific Borel $B$. (Theorem 6.43)


