A catalog of Cayley-Dickson-like products

TL;DR

This paper systematically catalogs 32 Cayley-Dickson-like products, identifying 8 that satisfy quaternion properties and representing each with a unique Fano Plane diagram.

Contribution

It provides a comprehensive classification of Cayley-Dickson-like doubling products and distinguishes the true Cayley-Dickson products from the others.

Findings

Identifies 8 true Cayley-Dickson doubling products

Provides Fano Plane representations for each product

Distinguishes products satisfying quaternion properties

Abstract

A catalog of all 32 Cayley-Dickson-like doubling products on ordered pairs (a,b),(c,d) for which (1,0) is the left and right identity and for which xx* and x*x equal the square of the norm of x given the conjugate (a,b)*=(a*,-b). Only eight of these are true Cayley-Dickson doubling products, since 24 of them do not satisfy the quaternion properties. Each of the eight Cayley-Dickson products has a distinctive representation in the Fano Plane.