From Special Relativity to embedded generators in Cartan subalgebras of rank-4 spin algebras

Starting by revisiting Special Relativity, here we provide a reliable characterization of the entire 4-dimensional fundamental structures in our reality where the frame of discrete tangent space of $F^{1,3}$ is quantized to massless, zero-momentum particles distributing on a 4-dimensional regular base $\{\mathbb{N}\cdot c_{\alpha }\}$ with metric $B(c_{\alpha },c_{\beta })={\delta }_{\alpha \beta }=\text{diag}(+,+,+,+)$, determining the constant $c$ locally, as well as instant characterizations on all particles moving along the proper time of $\tau \in \mathbb{N}$. Together with ${\varphi }_{\text{IV}}$ on 1-dimensional space $\{\mathbb{N}\cdot c_4\}$ of $B(c_{\alpha },c_4)=0$, the quantized particles of tangent frame are split anti-symmetrically from roots ${\gamma }_{\alpha 4}$ in a rank-4 Lie algebra with exactly $c_{\alpha }$ the generators of its Cartan subalgebra. As with the combined frame-Higgs valued in ${\gamma }_{\alpha 4}$, every massive particle is related to some split frame and Higgs to obtain its unique quantized relative 3-velocity, 4-velocity, and rest mass under the restriction of Pauli Exclusion Principle at each $\tau$. We calculated the domains $S_i\subset F$ and $S_0\subset F_{\ge }$ in $F_{\ge }\oplus F^3$ the spacetime in which a massive particle $\mathtt{p}$ evolves with $\{x^{\alpha }\}$ of $x^{\alpha }\in S_{\alpha }$ the coordinate its Center-of-Mass under an eligible observation that $\vert F\vert=\vert F_\geq\vert=\aleph_0<\vert$\mathbb{R}$\vert$, available for post-Newtonian approaches.