The abstract theory of Clifford modules was founded by a paper of M. F. Atiyah, R. Bott and Arnold S. Shapiro.
We will need to study anticommuting matrices (AB = −BA) because in Clifford algebras orthogonal vectors anticommute For the real Clifford algebra
One can always obtain another set of gamma matrices satisfying the same Clifford algebra by means of a similarity transformation.
Developed by Ettore Majorana, this Clifford module enables the construction of a Dirac-like equation without complex numbers, and its elements are called Majorana spinors.
The four basis vectors are the three Pauli matrices and a fourth antihermitian matrix.