Composition Algebras and Outer Automorphisms of Algebraic Groups

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 ${\mathbf{P}\mathbf{G}\mathbf{O}}^{+}(n)$ 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.