ABSTRACT
In this note, we establish an equivalence of categories between the category of all eight-dimensional composition algebras with any given quadratic form n over a field k of characteristic not two, and a category arising from an action of the projective similarity group of n on certain pairs of automorphisms of the group scheme PGO+(n) defined over k. This extends results recently obtained in the same direction for symmetric composition algebras. We also derive known results on composition algebras from our equivalence.
Notes
1Despite this notation, we shall not assume associativity, commutativity or unitality when we use the word algebra in general.
2Unless otherwise stated, all group actions are assumed to be from the left.