Classification of 7-dimensional subalgebras of 8-dimensional Clifford algebra

TL;DR

This paper classifies all 7-dimensional subalgebras within the 8-dimensional complex Clifford algebra, revealing exactly eight such subalgebras over the complex numbers and none over the reals.

Contribution

It provides a complete classification of 7-dimensional subalgebras of the 8-dimensional complex Clifford algebra using canonical bases.

Findings

Exactly eight 7-dimensional subalgebras over C identified

No 7-dimensional subalgebra exists over the reals

The identified subalgebras are contained in larger Clifford algebras over C

Abstract

All 7-dimensional subalgebras of the 8-dimensional Clifford algebra over the field C of complex numbers are found. Canonical bases are used throughout the determination. It is found that the 8-dimensional Clifford algebra over C has exactly eight 7-dimensional subalgebras and each of these eight is over the complex numbers. It follows that the 8-dimensional Clifford algebra over C has no 7-dimensional subalgebra over the real numbers. Another consequence is that the eight 7-dimensional subalgebras of the 8-dimentional Clifford over C are in fact 7-dimensional subalgebras of all greater dimensional Clifford algebras over C.