Relative position of Borel subgroups
= Relative position of Borel subgroups
The <relative position of Borel subgroups> $B_1,B_2$ is the <Weyl group> index of their simultaneous conjugacy orbit. After moving the first subgroup to a fixed $B$, the second has the form $gBg^{-1}$ and its index is the double coset $BgB$. This identifies the diagonal action on pairs of flags with the <Bruhat decomposition>.