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 G be a simply connected absolutely simple group of outer type An or of type Bn, Cn or Dn (non trialitarian) defined over an arbitrary field F. If the separable dimension of F is at most 2 for every torsion prime of G, then every G-torsor is trivial.