In this paper, the Dirac projection operators for energy, helicity, chirality, and spin have been generalized to a quaternionic representation. We present explicit, step-by-step algebraic checks of the projector action on two-complex-component Dirac spinors 1, demonstrating how the quaternionic structure consistently reproduces the expected eigenvalue relations and projection properties. Furthermore, the non-commutative nature of the quaternion algebra is carefully handled to ensure the idempotency, orthogonality, and completeness of the generalized projectors. This formulation not only preserves the essential physical content of the standard Dirac theory but also provides a more unified and potentially richer framework for analyzing spinor dynamics and internal symmetries.
Singh et al. (Fri,) studied this question.