Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Jean-Pierre Gazeau

TL;DR

The paper introduces a concise $$(R+L)$$ mnemonic for computing products of (split) octonions expressed in Cayley–Dickson form $q+\ell p$, recasting the multiplication as a 2×2 quaternionic block pattern in which half the elementary products appear in reversed order. This structure is formalized via an interlaced 2×2 matrix product and a reversal marker, with a map $M(o)=q\ell^2 p^{*} p q^{*}$ encoding the construction; the rule reproduces Cayley–Dickson multiplication and extends verbatim through the entire Cayley–Dickson tower. The interlaced product is generalized to matrices over a unital non-associative algebra, and the work connects to matrix- and geometrically flavored frameworks associated with exceptional structures, including potential norm-preserving groups and determinant-like invariants. In addition to offering a practical computation tool, the paper suggests structured matrix formalisms and algebraic identities relevant to non-associative algebras and their role in exceptional geometry and Jordan-algebra contexts.

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.

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

TL;DR

The paper introduces a concise mnemonic for computing products of (split) octonions expressed in Cayley–Dickson form , recasting the multiplication as a 2×2 quaternionic block pattern in which half the elementary products appear in reversed order. This structure is formalized via an interlaced 2×2 matrix product and a reversal marker, with a map encoding the construction; the rule reproduces Cayley–Dickson multiplication and extends verbatim through the entire Cayley–Dickson tower. The interlaced product is generalized to matrices over a unital non-associative algebra, and the work connects to matrix- and geometrically flavored frameworks associated with exceptional structures, including potential norm-preserving groups and determinant-like invariants. In addition to offering a practical computation tool, the paper suggests structured matrix formalisms and algebraic identities relevant to non-associative algebras and their role in exceptional geometry and Jordan-algebra contexts.

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.
Paper Structure (7 sections, 3 theorems, 11 equations)

This paper contains 7 sections, 3 theorems, 11 equations.

Table of Contents

  1. Introduction
  2. Cayley--Dickson form and a $2\times2$ quaternionic model
  3. The mnemonic $(R+L)$ pattern
  4. Propagation through the Cayley--Dickson tower
  5. Toward matrix structures over non-associative algebras
  6. Interlaced $2\times2$ multiplication over a unital non-associative algebra
  7. Conclusion

Key Result

Proposition 2.4

For $o_i=q_i+\ell p_i$ with $q_i,p_i\in\mathbb{H}$, where $o_1o_2$ is given by eq:CD.

Theorems & Definitions (11)

  • Definition 2.1: Reversal marker
  • Remark 2.2
  • Definition 2.3: Interlaced $2\times2$ product over $\mathbb{H}$
  • Proposition 2.4: Matrix mnemonic for octonion multiplication
  • Remark 2.5
  • Remark 2.6
  • Theorem 4.1: $(R+L)$ propagation
  • Remark 4.2
  • Remark 6.1: On inverses and "determinants"
  • Proposition 6.2: One-sided inverses under nucleus/centrality assumptions
  • ...and 1 more