  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 $\mathbf{PGO}^+(n)$ 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.
