Product of three octonions

TL;DR

This paper explores the algebraic structure of octonions by decomposing the product of three octonions into orthogonal components, generalizing concepts like the cross product, and relates it to known solutions.

Contribution

It introduces a straightforward decomposition of three-octonion products into orthogonal parts, linking it to existing complex algebraic solutions.

Findings

Decomposition into anticommutator, commutator, and associator components.

Generalization of the cross product to three arguments for octonions.

Equivalence to Okubo's known solution.

Abstract

This paper is devoted to octonions that are the eight-dimensional hypercomplex numbers characterized by multiplicative non-associativity. The decomposition of the product of three octonions with the conjugated central factor into the sum of mutually orthogonal anticommutator, commutator and associator, is introduced in an obvious way by commuting of factors and alternating the multiplication order. The commutator is regarded as a generalization of the cross product to the case of three arguments both for quaternions and for octonions. It is verified that the resulting additive decomposition is equivalent to the known solution derived and presented by S. Okubo in a cumbersome form.