Galois cohomology of the classical groups over imperfect fields

Elaborating on techniques of Bayer-Fluckiger and Parimala, we prove the following strong version of Serre’s Conjecture II for classical groups: let GG be a simply connected absolutely simple group of outer type AnAn or of type BnBn, CnCn or DnDn (non trialitarian) defined over an arbitrary field FF. If the separable dimension of FF is at most 2 for every torsion prime of GG, then every GG-torsor is trivial.