Linear operators in Clifford algebras (Q1184567)

It is well known that one way to construct spinors is to define them as elements of minimal left (or right) ideals in Clifford algebras. These ideals are then spinor spaces, and linear operators in those spaces are particular linear endomorphisms of Clifford algebras. The author considers the real vector space structure of the algebra of linear endomorphisms for a finite-dimensional real Clifford algebra (2,4,5,6,7,8). A basis of that space is constructed in terms of the operators \(M_{eI,eJ}\in\text{End}_{\mathbb{R}}(Cl(V))\) defined by \(x\mapsto e_ Ixe_ J\), where the \(e_ I\) are the generators of the Clifford algebra and \(I\) is a multi-index (3,7). In particular, it is shown that the family \((M_{eI,eJ})\) is exactly a basis in the even case.