A parabolic \(F\)-subgroup of a reductive group is of the form \(P(\lambda)\) for an \(F\)-defined cocharacter \(\lambda:\mathbb{G}_m\to G\). The group \[ Z_G(\lambda)=C_G(\operatorname{im}\lambda) \] is a connected reductive \(F\)-subgroup, while the contracting subgroup \[ U(\lambda)=\{g:\lim_{t\to 0}\lambda(t)g\lambda(t)^{-1}=e\} \] is connected unipotent and \(F\)-split. The product map \[ Z_G(\lambda)\ltimes U(\lambda)\to P(\lambda) \] is an isomorphism of varieties and groups, so \(U(\lambda)=R_u(P)\) and \(Z_G(\lambda)\) is an \(F\)-Levi subgroup.

If \(L\) is another \(F\)-defined Levi subgroup, it is the centralizer of an \(F\)-defined torus in \(P\). Geometric Levi conjugacy gives a unique element of \(R_u(P)\) carrying \(Z_G(\lambda)\) to \(L\). The transporter is \(F\)-defined and consists of one point, hence that point lies in \(R_u(P)(F)\). The maximal \(F\)-split torus statement follows by choosing a minimal parabolic \(F\)-subgroup inside \(P\) and applying the conjugacy of maximal \(F\)-split tori.