Geometric Construction of Lorentz Boosts: Linear Strokes and their Quaternionic Duals

[Work in Progress (V1)] Abstract: This paper explores a biquaternionic approach to relativistic kinematics, serving as a dual framework to the non-linear, real-quaternionic representation developed in the author's previous works [HL14, HL15, HL26]. Building upon the standard biquaternionic formulation of Lorentz transformations, we introduce the concept of Linear Strokes --direct analogues to the involutive Stroke operators introduced in [HL26]. This bridging concept allows for a natural translation of several key results between the two domains. Specifically, we demonstrate the geometric construction of Lorentz boosts and relativistic velocity addition using these new building blocks. Furthermore, core constructs such as the Zorro Signature, Stroke Reduction Lemmas, and the Four-Stroke Theorem are shown to possess direct dual equivalents. Finally, we remark that such geometric concepts and results naturally extend to the (non-associative) Algebra of Octonions.