The question is asked in the context of (connected) reductive groups.
In the article i'm working on, the author states the following fact (well it's not word to word exact, I simplified it a little) :
If we choose a parabolic subgroup, determined by a simple reflection $s$ in $W$ (the Weyl group, given a maximal torus), then the variety of the Borel subgroups contained in $P$ is one-dimensional, moreover it can be identified with the projective line.
Is there a simple way to prove this ?
What I tried : since the parabolic subgroup $P$ is determined by a simple reflection, I wrote down the Levi decomposition using roots, then I was thinking that the Borel subgroups contained in $P$ are in bijection with the Borel subgroups of the Levi complement.