Let $G$ be a connected, reductive group over an algebraically closed field $k$. Let $T$ be a maximal torus of $G$, and $\Phi = \Phi(T,G)$ the set of roots of $T$ in $G$. The bases $\Delta$ of $\Phi$ are parameterized by the Borel subgroups of $G$ containing $T$.
Now assume that $k$ is not necessarily algebraically closed. Replace $T$ by a maximal $k$-split torus of $G$. Is there an analogous parameterization for the bases of $\Phi$? For example, are minimal $k$-parabolic subgroups containing $T$ in one to one correspondence with such bases?
My guess would be yes: if $P_0$ is a minimal parabolic, then $P_0$ contains $M_0 = Z_G(A_0)$, then I suppose $\mathscr R_u(P_0)$ comprises a system of positive roots. Conversely, I imagine relative root subgroups of a give system of positive roots $\Phi^+$ would comprise a connected unipotent subgroup $N_0$ for which $P_0 = M_0N_0$ is a minimal $k$-parabolic subgroup, for which we recover $N_0 = \mathscr R_u(P_0)$.