 Bruhat decomposition of SL2 over a finite field

This is the finite special-linear-group instance of the general Bruhat decomposition.

For the upper-triangular subgroup $B$ of $S{L}_2({\mathbb{F}}_p)$ andthe two double cosets are $B$ and $BwB$, and $B\cap wB{w}^{-1}=T$, the diagonal subgroup.