Triality over Schemes

Working over an arbitrary base scheme, we provide an alternative development of triality which does not use Octonion algebras or symmetric composition algebras. Instead, we use the Clifford algebra of the split hyperbolic quadratic form of rank 8 and computations with Chevalley generators of groups of type D₄. Following the strategy of The Book of Involutions KMRT, we then define the stack of trialitarian triples and show it is equivalent to the gerbe of PGO₈^+--torsors. We show it has endomorphisms generating a group isomorphic to S₃ and that several familiar cohomological properties of PGO₈^+ follow in this setting as a result. Next, we define the stack of trialitarian algebras and show it is equivalent to the gerbe of PGO₈^+ S₃--torsors. Because of this, it is also equivalent to the gerbes of simply connected, respectively adjoint, groups of type D₄. We define SpinT and PGO^+T for a trialitarian algebra and define concrete functors T SpinT and T PGO^+T which realize these equivalences.