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In this article, we bypass the detailed symmetry breaking pathwaysestablished in [1]. Instead, a direct route from the Spin(10) model to theStandard Model is enabled via a single algebraic constraint. This singleconstraint, however, may be reconfigured as a requirement that three$\mathfrak{so}(10)$ actions coincide on a fixed space of multi-vector fermions.This $\mathfrak{so}(10) \mapsto \mathfrak{su}(3)_{C} \oplus\mathfrak{su}(2)_{L} \oplus{u}(1)_{Y}$ breaking (from a three-way intersection)mirrors, in certain ways, the $\mathfrak{so}(8) \mapsto \mathfrak{g}_2$breaking (from a three-way intersection) in the context of octonionic triality.By extending this result to include quaternions and complex numbers, we findthat a five-way intersection breaks $\mathfrak{so}(10) \mapsto\mathfrak{su}(3)_{C} \oplus \mathfrak{u}(1)_{Q}$. These are the StandardModel's unbroken gauge symmetries, post-Higgs-mechanism.

An Algebraic Roadmap of Particle Theories, Part III: Intersections