  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.
