Split rank of inner forms

Every torus in $G$ transfers to $G^*$, so we definitely have the desired inequality. If we have equality, then there is a maximal split torus $A$ in $G$ that is also maximal split when transferred to the torus $A^*$ in $G^*$; so $C_{G^*}(A^*)$ is a torus; so $C_G(A)$, which is isomorphic over the separable closure to $C_{G^*}(A^*)$, is a torus; so $G$ is quasisplit, hence isomorphic to $G^*$.