An Algebraic Roadmap of Particle Theories, Part I: General construction

Expanding the results of [1], [2], [3], we demonstrate a network of algebraic connections between six well-known particle theories. These are the Spin(10) model, the Georgi-Glashow model, the Pati-Salam model, the Left-Right Symmetric model, the Standard Model both pre- and post-Higgs mechanism. A new inclusion of a quaternionic reflection within the network further differentiates $W^{\pm}$ bosons from the $Z^0$ boson in comparison to the Standard Model. It may introduce subtle new considerations for the phenomenology of electroweak symmetry breaking.