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

This paper introduces a novel, compact "(R+L)" mnemonic rule for efficiently computing products in (split) octonions and extends this matrix-based pattern to all algebras within the Cayley--Dickson tower, filling a gap in the classical literature on non-associative composition algebras.

Original authors: Jean-Pierre Gazeau

Published 2026-01-27

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Jean-Pierre Gazeau

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to bake a very complex cake, but the recipe is written in a language that keeps changing the order of your ingredients. Sometimes you mix flour and sugar, but the recipe says, "Actually, mix sugar and flour, then add eggs, but remember to flip the eggs first." This is what mathematicians face when working with Octonions.

Octonions are a type of number system that extends the familiar real numbers, complex numbers, and quaternions. They are incredibly useful in advanced physics and geometry, but they are notoriously difficult to use because they are non-associative. In simple terms, this means the order in which you multiply things matters immensely, and the rules get messy very quickly.

Here is what Jean-Pierre Gazeau's paper does to make this easier:

1. The Problem: A Messy Recipe

The standard way to multiply two Octonions involves a formula that mixes normal multiplication with "conjugation" (a bit like flipping a sign) and strict rules about which number comes first. It's like a recipe that says: "Take the first part of the cake, multiply it by the second part, but then take the second part of the first cake, flip it, and multiply it by the first part of the second cake." It's easy to get lost and make a mistake.

2. The Solution: A "Left-Right" Pattern

The author introduces a clever trick, or a mnemonic (a memory aid), to organize this chaos. He suggests writing the Octonion not as a single messy formula, but as a 2x2 grid (like a tic-tac-toe board) made of simpler numbers called Quaternions.

Once you put the numbers in this grid, the multiplication rule becomes a beautiful, repeating pattern. The author calls it the "(R + L)" pattern:

R stands for Right (the natural order: A times B).
L stands for Left (the reversed order: B times A).

The magic is that every single box in your 2x2 grid is just the sum of two things: one calculated normally (R) and one calculated backward (L). The pattern looks like this:

Top-Left: Right + Left
Top-Right: Left + Right
Bottom-Left: Left + Right
Bottom-Right: Right + Left

It's like a dance where half the dancers move forward and half move backward, but they always end up in the same perfect formation. You don't have to remember why the order flips; you just follow the pattern: R+L, L+R, L+R, R+L.

3. The "Interlaced" Multiplication

To make this work, the author invents a special way to multiply these 2x2 grids, which he calls "interlaced" multiplication.

In normal matrix multiplication, you just multiply rows and columns.
In this new "interlaced" method, when you multiply the "off-diagonal" parts (the corners that aren't on the main line), you intentionally swap the order of the numbers.

This swap is the secret sauce that makes the math work for Octonions. It's like a special handshake where you have to switch hands before you shake.

4. The Tower of Babel (The Cayley-Dickson Tower)

The paper claims this trick isn't just for Octonions. There is a whole "tower" of number systems built on top of each other (starting from real numbers, going to complex, then quaternions, then octonions, and beyond).

The author shows that this same R+L pattern works for every level of this tower.
No matter how high you climb in this tower of numbers, you can always use this same 2x2 grid and the same "swap the order" rule to do your math.

5. Why This Matters (According to the Paper)

The paper doesn't claim this will cure diseases or build new bridges immediately. Instead, it offers two main benefits:

Pedagogy (Teaching): It gives students and researchers a much simpler, visual way to learn and teach these difficult numbers. Instead of memorizing a confusing formula, they just remember the "Right + Left" dance pattern.
Computation: It provides a reliable tool for computers or humans to calculate these products without making errors about which order to multiply in.

The author also hints that this "interlaced" multiplication could help mathematicians build new types of matrix rules for these weird, non-associative number systems, potentially helping them understand complex shapes in geometry (like those related to the "exceptional Lie groups" mentioned in the text).

In a nutshell: The paper takes a confusing, order-sensitive math problem and turns it into a simple, repeating pattern of "forward and backward" steps, making it much easier to handle these exotic numbers.

Technical Summary: A Mnemonic Matrix Rule for (Split) Octonionic Multiplication and its Extension to the Cayley–Dickson Tower

Problem Statement
The multiplication of octonions, typically constructed via the Cayley–Dickson process as pairs of quaternions $(q, p) \in \mathbb{H} \oplus \mathbb{H}\ell$ , is governed by the formula:
$(q_1 + \ell p_1)(q_2 + \ell p_2) = q_1q_2 + \ell^2 p_2p_1^* + \ell(q_2p_1 + q_1^*p_2)$
where $\ell^2 = \pm 1$ distinguishes between division and split octonions, and $*$ denotes quaternionic conjugation. While standard, this formulation is computationally cumbersome because it intermixes conjugation with order-sensitive products. Direct application requires careful tracking of which terms are reversed and where conjugations apply, a process prone to error, particularly when extending these operations to higher-dimensional algebras in the Cayley–Dickson tower.

Methodology
The paper introduces a compact mnemonic device that reorganizes octonionic multiplication into a $2 \times 2$ block matrix model. The core methodology involves:

Matrix Representation: Defining a linear map $M$ that embeds an octonion $o = q + \ell p$ into a $2 \times 2$ matrix over quaternions:
$M(o) = \begin{pmatrix} q & \ell^2 p^* \\ p & q^* \end{pmatrix}$
Interlaced Product: Defining a specialized multiplication rule, denoted $\langle \rangle$ $⟨ ⟩$ , for these matrices. Unlike standard matrix multiplication, this rule applies a systematic "order reversal" to specific off-diagonal contractions. For two matrices $A$ $A$ and $B$ $B$ , the product $A \langle \rangle B$ $A ⟨ ⟩ B$ is computed such that:
- Diagonal blocks are sums of a right-ordered product and a left-ordered (reversed) product.
- Off-diagonal blocks similarly mix orders, specifically reversing the multiplication order for terms involving the "cross" interactions (e.g., $bc'$ becomes $c'b$ ).
Generalization: Extending this construction to the entire Cayley–Dickson tower ( $A_n$ ), representing elements of $A_{n+1}$ as pairs $(a, b)$ with $a, b \in A_n$ , and applying the same interlaced product rule to the resulting $2 \times 2$ block matrices over $A_n$ .

Key Contributions and Results

The (R + L) Mnemonic Rule: The primary result is the identification of a universal pattern for the product of two elements in Cayley–Dickson form. Every block in the resulting $2 \times 2$ matrix is the sum of exactly two terms: one in natural (right) order ( $R$ ) and one in reversed (left) order ( $L$ ). The pattern is explicitly:
$\begin{pmatrix} R + L & L + R \\ L + R & R + L \end{pmatrix}$
This rule allows for the computation of octonionic (and split octonionic) products without repeatedly re-deriving conjugation and order-reversal rules.
Propagation to the Cayley–Dickson Tower: The paper proves (Theorem 4.1) that this interlaced product reproduces the Cayley–Dickson multiplication for all algebras $A_n$ of dimension $2^n$ . The (R + L) pattern propagates inductively, providing a unified computational tool for non-associative algebras beyond the octonions.
Matrix Structures over Non-Associative Algebras: The author defines an "interlaced" product for $2 \times 2$ matrices over any unital non-associative algebra. The paper establishes conditions under which this structure supports one-sided inverses (Proposition 6.2), specifically requiring that the "determinant-like" terms ( $\Delta_L = da - cb$ and $\Delta_R = ad - cb$ ) belong to the left and right nuclei of the algebra, respectively.

Significance and Claims
The paper claims that this explicit "(R + L)" mnemonic does not appear in the classical literature on octonions or composition algebras. Its significance lies in:

Pedagogical Utility: It offers a visual and systematic way to handle the non-associativity of octonions by shifting the complexity from ad-hoc conjugation rules to a fixed pattern of order reversals.
Computational Efficiency: It provides a reliable method for computing products in Cayley–Dickson form, reducing the cognitive load of tracking order sensitivity.
Theoretical Bridge: The construction serves as a bridge between the non-associative world of the Cayley–Dickson sequence and classical matrix formalisms. It suggests that structured "matrix-like" formalisms can be adapted for non-associative settings, potentially aiding in the study of exceptional geometry, Jordan algebras (such as the Albert algebra), and the triality phenomena associated with $\text{Spin}(8)$ .

The author notes that while the mnemonic remains valid as an identity for the defining multiplication even in algebras with zero divisors (where $n \ge 4$ ), care must be taken regarding invertibility and determinant-like constructions in these non-alternative regimes.