This paper proves, by explicit Clifford-algebra computation, that the γ0-diagonal complete orthogonal primitive decomposition of real Cl(1,3) has two sectors, while the extension to Cl(1,4) supplies a second independent commuting basis-blade involution, γ1234. Because I5 = γ0γ1234, the generated involution group is Z2 × Z2, not (Z2)3. The resulting decomposition contains four primitive idempotents, each generating a real 8-dimensional minimal left ideal. The count four is itself fixed by Wedderburn theory, since each simple summand has rank two; the explicit content of the result is the basis-blade realization of the family and the demonstration of why a naive three-involution count would predict eight. The central idempotents (1 ± I5)/2 realize the real semisimple splitting Cl(1,4) ≅ M2(H) ⊕ M2(H), where H denotes the quaternions. They do not, by themselves, supply a complex imaginary unit; a complex spinor interpretation still requires choosing an internal quaternionic complex structure within each ideal. Physical identifications with chirality, charge, helicity, or U(1) phase are therefore treated as conjectural.Supporting results include: (i) a Pancharatnam-phase computation from projector triple products in Cl(3,0), confirming the half-solid-angle rule 2|φ| = Ω for geodesic spherical triangles; (ii) a bivector-square classification of geometric-i candidates, with the observation that the chiral split in real Cl(1,3) is a complex-structure operation, not a real idempotent split; and (iii) a short positioning of the result relative to real Cl(0,6) and complex Clu2086 Standard Model programs. The paper is intended as a lower-dimensional real-idempotent audit, not as a derivation of a Standard Model generation.