Isotropy of symplectic involutions and motivic equivalence for groups of type $\mathsf{C}$

We extend to arbitrary characteristic a theorem due to Karpenko and Tignol on the isotropy of symplectic involutions after generic index reduction; as a consequence, we also prove the existence of critical varieties for semi-simple algebraic groups of type $\mathsf{C}$ over arbitrary base fields. The latter result relies on a characteristic-free argument, which involves the computation of the mod $2$ Tate trace of the variety of isotropic ideals of reduced dimension $2$ for a symplectic involution.