The gauge symmetries and fermionic sector of the Standard Model are derived from the free Dirac Lagrangian in 8-dimensional spacetime. Motivated by second quantization, we embed the 4-dimensional Clifford algebra of the Dirac Lagrangian into that of 8-dimensional spacetime, showing this embedding carries a redundancy described by a complexified Pati-Salam group acting on field multiplets represented as Clifford algebra ideals, externally and internally via left and right multiplication. Invariance of the Dirac Lagrangian selects $SU(4)_C \times SU(2)_L \times SU(2)_R$ internally and $SU(4)_F \times Spin(1,3)$ externally. Fixing a triality in $C\ell_{8,0}$ simultaneously gives $SU(4)_C \to SU(3)_C \times U(1)_{B-L}$ and $SU(4)_F \to SU(3)_F \times U(1)_{F}$, originating a leptonic sector and a non-mixing fourth family. Breaking $U(1)_{F}$ yields a hierarchy governed by a generalized Koide formula, with mass scales showing a modular nature.