A Mnemonic Matrix Rule for (Split) Octonionic Multiplication and its Extension to the Cayley--Dickson Tower

TL;DR

This paper introduces a simple mnemonic rule for computing products of (split) octonions using a quaternionic matrix pattern, which extends to all Cayley--Dickson algebras, aiding efficient calculations in non-associative algebraic structures.

Contribution

The paper presents a novel, explicit (R+L) mnemonic rule for octonionic multiplication that extends to the entire Cayley--Dickson tower, not previously documented in literature.

Findings

Provides a compact computational rule for octonions.

Extends the rule to all Cayley--Dickson algebras.

Facilitates efficient calculations in non-associative algebras.

Abstract

We present a compact mnemonic device for computing the product of two (split) octonions written in Cayley--Dickson form q+l p with q,p in H. The rule appears as a simple (R+L) pattern of right-ordered and left-ordered (quaternionic) products inside a 2X2 quaternionic matrix model. The pattern extends verbatim to all algebras in the Cayley--Dickson tower, providing an efficient computational tool in non-associative settings. To our knowledge, this explicit ``(R+L)'' mnemonic does not appear in the classical literature on octonions or composition algebras.