On the Octonion-like Associative Division Algebra

TL;DR

This paper clarifies the structure of an octonion-like algebra, showing it is equivalent to the split-biquaternion algebra and establishing its properties as a seminormed division algebra over the real numbers.

Contribution

It demonstrates that the octonion-like algebra is identical to the split-biquaternion algebra and provides elementary linear algebra methods for its analysis and implementation.

Findings

The algebra is the same as split-biquaternion algebra.

It is a seminormed division algebra over R.

Elementary linear algebra facilitates software implementation.

Abstract

Using elementary linear algebra, this paper clarifies and proves some concepts about a recently introduced octonion-like associative division algebra over R. This octonion-like algebra is actually the same as the split-biquaternion algebra, an even subalgebra of Clifford algebra Cl(4,0). For two seminorms described in the paper, it is shown that the octonion-like algebra is a seminormed composition algebra over R. Moreover, additional results related to the computation of inverse numbers in the octonion-like algebra are introduced, showing that the octonion-like algebra is a seminormed division algebra over R, i.e., division by any number is possible as long as the two seminorms are non-zero. Additional results on normalization of octonion-like numbers and some involutions are also presented. The elementary linear algebra descriptions used in the paper also allow straightforward software implementations of the octonion-like algebra.