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

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.