  Trialitarian triples are triples of central simple algebras of degree 8 with
orthogonal involution that provide a convenient structure for the
representation of trialitarian algebraic groups as automorphism groups. This
paper explicitly describes the canonical "trialitarian'' isomorphisms between
the spin groups of the algebras with involution involved in a trialitarian
triple, using a rationally defined shift operator that cyclically permutes the
algebras. The construction relies on compositions of quadratic spaces of
dimension 8, which yield all the trialitarian triples of split algebras. No
restriction on the characteristic of the base field is needed.
