An Algebraic Description of the Exceptional Isogenies to Orthogonal Groups

We show how the exceptional isogenies of classical groups to orthogonal groups of quadratic spaces of dimensions up to 8 over fields of characteristic different from 2 may be obtained by explicit algebraic constructions using the Cartan--Dieudonn\'{e} Theorem. This gives a unified approach to groups arising from quaternion algebras, bi-quaternion algebras, and matrices over bi-quaternion algebras. The latter seem to yield new isogenies, which are related to triality in the hyperbolic case.