Trialitarian groups and the Hasse principle

Let F be a field of characteristic ≠ 2 such that is of cohomological 2- and 3-dimension ≤ 2. For G a simply connected group of type ³ D ₄ or ⁶ D ₄ over F, we show that the natural map where Ω _F is the set of orderings of F and F _v denotes the completion of F at v, restricts to be injective on the image of H ¹(F, Z(G)) in H ¹(F, G). For F not formally real, this implies that Serre's “Conjecture II” [Ser.94,III.3.1] holds for such groups if and only if trialitarian groups are classified by their Tits algebras over F. Received: 17 September 1998