If \(G\) has a proper pseudo-parabolic \(F\)-subgroup \(P\), the \(F\)-rational fixed point theorem for \(G/P\) shows that each maximal \(F\)-split torus is \(G(F)\)-conjugate into \(P\). The result then follows by induction inside \(P\).

If no proper pseudo-parabolic \(F\)-subgroup exists, Springer's criterion forces all \(F\)-split subtori to lie in the solvable \(F\)-radical. The conjugacy statement is then the corresponding conjugacy theorem for maximal \(F\)-split tori in connected solvable groups, where the unipotent radical acts transitively on the choices. This completes the induction.