Well, Papa, can you multiply triplets?

TL;DR

This paper explores the algebraic structures of quaternions and octonions, revealing that quaternions can be viewed as a commutative graded algebra, while octonions cannot, highlighting fundamental differences.

Contribution

It provides a novel interpretation of quaternion algebra as a commutative graded algebra and demonstrates the impossibility of a similar interpretation for octonions.

Findings

Quaternions form a commutative -graded algebra.

Octonions cannot be interpreted as a similar graded algebra.

The algebraic structures of quaternions and octonions differ fundamentally.

Abstract

We show that the classical algebra of quaternions is a commutative $\Z_2\times\Z_2\times\Z_2$-graded algebra. A similar interpretation of the algebra of octonions is impossible.