Biquaternion Construction of the Lorentz Transformation Operator in Relativistic Quantum Mechanics

The Lorentz transformation properties of the Dirac matrices remain a source of confusion in relativistic quantum mechanics. In standard presentations the covariance of the Dirac equation is guaranteed by assuming the existence of a transformation operator $S$ satisfying $S\gamma^\nu S^{-1}=\Lambda_\mu^{~\nu}\gamma^\mu$, and then solving for $S$. In this paper I take a constructive approach: starting from a biquaternion Pauli-level formalism that is morphologically equivalent to Minkowski space-time, I build the Weyl and Dirac matrix environments explicitly. By doubling the Pauli-level basis into a PT-duplex structure, the Weyl and Dirac $\beta_\mu$ matrices arise naturally as Clifford four-sets connected directly to space-time. From this basis, Lorentz transformation operators are constructed step by step, showing that the operator $S\Lambda S^{-1}$ follows from the algebra rather than being postulated. This approach clarifies the apparent contradictions regarding whether Dirac matrices transform as four-vectors or remain inert, by demonstrating that in the Feynman slash objects ${\fs P}=P_\mu\beta^\mu$ one may transform either the coordinates $P_\mu$ or the metric $\beta^\mu$, but not both simultaneously. Spinor transformation rules and the mixing of Pauli spinors in the Dirac representation are derived constructively, revealing why the Pauli equation cannot be relativistic on its own. The results provide a transparent biquaternion framework for the Lorentz covariance of relativistic wave equations, and may serve as a methodological clarification of the algebraic foundations of relativistic quantum mechanics.