Quaternionic Möbius Transformations: a Different Approach to Relativistic Kinematics (Work in Progress).

[Work in Progress - Version 3] Abstract: As the most general finite-dimensional associative division algebra over the reals, quaternions possess a natural 1+3 dimensional structure. When exploring their deep mathematical properties —such as Möbius transformations— the fundamental laws of Special Relativity, including the Lorentz group, emerge with striking elegance. In this paper, we develop a non-linear geometric approach to relativistic kinematics by studying transformations of the form q -> (aq-b)/(bq+a) [right division]. We demonstrate that the action of this group is conformal on the sphere of square roots of -1 (denoted as L), providing a robust foundation for relativistic boosts and rotations. By algebraically resolving the fixed points of these transformations, we uncover highly intuitive visualizations of frame transitions and kinematic effects, including a geometric representation of the twin paradox. Building upon my earlier research, we establish a rigorous isomorphism with the standard SO+(1,3) formulation and introduce a unified set of equivalent velocity addition formulas. A central algebraic contribution is the formalization of the "Forgotten Fundamental Formula": a surprisingly simple expression taking the form of a multidimensional tangent addition: V*W = (V+W)/(1-VW) [right division]. Furthermore, this framework bridges the gap between abstract algebra and pure geometry by detailing novel, highly accessible compass and straightedge constructions for relativistic velocity composition. Key updates in Version 3: This version (approx. 146 pages) introduces the preliminary draft of Section 4.3 (Geometry), featuring novel straightedge and compass constructions for relativistic velocity addition. Includes the formalization of the "Forgotten Fundamental Formula" and its operational hybrid variants (Section 4.2). Disclaimer: This is a working paper and a work in progress. It contains unfinished sections (notably 4.3.4 and onwards), placeholders, and may contain typos or notation inconsistencies. The structure and content are subject to further refinement before the final publication.