# Eight-dimensional Octonion-like but Associative Normed Division Algebra

## TL;DR

This paper introduces an eight-dimensional associative algebra resembling octonions, which maintains the norm composition law and allows for a parallelizable 7-sphere with different topology from traditional octonions.

## Contribution

It constructs an eight-dimensional associative algebra that mimics octonion properties, including norm preservation and sphere parallelizability, with a novel geometric product-based norm.

## Key findings

- Defines an eight-dimensional associative algebra with octonion-like properties.
- Shows the algebra respects the norm composition law using geometric product.
- Demonstrates the 7-sphere is parallelizable within this algebra, with distinct topology.

## Abstract

We present an eight-dimensional even sub-algebra of the ${2^4=16}$-dimensional associative Clifford algebra ${\mathrm{Cl}_{4,0}}$ and show that its eight-dimensional multivectors ${\bf X}$ and ${\bf Y}$ respect the composition law ${||{\bf X}{\bf Y}||=||{\bf X}||\,||{\bf Y}||}$, thus forming an octonion-like but associative normed division algebra, where the norms are calculated using the fundamental geometric product instead of the usual scalar product so that the underlying coefficient algebra resembles split complex numbers instead of reals. The corresponding 7-sphere obtained from projecting this multivector-valued composition law to the scalar-valued composition law has a topology that differs from that of the octonionic 7-sphere. Just as the octonionic 7-sphere is parallelizable using the non-associative algebra of octonions, we demonstrate that the 7-sphere presented herein is parallelizable using the said associative algebra.