The Alternative Clifford Algebra of a Ternary Quadratic Form

TL;DR

This paper demonstrates that the alternative Clifford algebra associated with a nondegenerate ternary quadratic form is isomorphic to an octonion algebra over a polynomial ring, revealing a deep algebraic structure.

Contribution

It establishes a novel connection between alternative Clifford algebras of ternary quadratic forms and octonion algebras over polynomial rings.

Findings

Alternative Clifford algebra is an octonion algebra

The result holds over polynomial rings in one variable

Provides new insights into algebraic structures of quadratic forms

Abstract

We prove that the alternative Clifford algebra of a nondegenerate ternary quadratic form is an octonion algebra over the ring of polynomials in one variable over the field of definition.