Gauge transformations of spinors within a Clifford algebraic structure

Algebraic spinors can be defined as minimal left ideals of Clifford algebras. We consider gauge transformations which an two-sided equivalence transformations of a complete algebra, including the spinors. These transformations of the spinors introduce new interaction terms which appear hard to interpret. We establish algebraic theorems which allow these new interaction terms to be evaluated and use these ideas to provide a new formulation of Glashow's electroweak interactions of leptons. The theorems also lead us to propose a new Clifford algebraic definition of spinors based on nilpotents, rather than idempotents.