An octonion algebra originating in combinatorics

TL;DR

This paper explores new polynomial identities related to octonion algebra originating from combinatorics, showing their equivalence and connecting them to classical sum of squares identities.

Contribution

It demonstrates that all polynomial Lagrange identities derived from combinatorics are fundamentally equivalent, extending the classical sum of squares identities to octonion algebra.

Findings

All new identities are equivalent to each other.

Connections established between combinatorial identities and octonion algebra.

Extension of classical sum of squares identities to polynomial forms.

Abstract

C.H. Yang discovered a polynomial version of the classical Lagrange identity expressing the product of two sums of four squares as another sum of four squares. He used it to give short proofs of some important theorems on composition of delta-codes (now known as T-sequences). We investigate the possible new versions of his polynomial Lagrange identity. Our main result shows that all such identities are equivalent to each other.