Upgrading Tensors

The covariant derivative of the standard model has 3 parts: a coupling strength, a group generator, and a potential. Constrained to 4 dimensions, tensors can be equipped with multiplication and division via an isomorphism to quaternions. U(1) symmetry is then a normalized quaternion, SU(2) a unit quaternion, and SU(3) the product of two normalized quaternions. One tensor can house the 3 symmetries of the standard model: D$_{\mu} \rightarrow (\frac{A}{|A|}exp(A-A^*))^*\frac{B}{|B|}exp(B-B^*) \nabla$.