We construct a number system that is isomorphic to the factor ring of an arbitrary polynomial. If this polynomial is the minimal polynomial of a given linear operator, then its generalized spectral decomposition is immediately determined. The same methods apply to a linear operator over a finite field. Clifford algebra arises when we introduce a grading onto the algebra of endomorphisms.
Sobczyk, G. (1996). Structure of factor algebras and Clifford algebra. Linear Algebra and Its Applications, 241–243, 803–810. https://doi.org/10.1016/0024-3795(95)00604-4