Three generations and a trio of trialities

We identify the Standard Model's su(3)⊕su(2)⊕u(1) internal symmetries within the triality symmetries tri(C)⊕tri(H)⊕tri(O). From here, the corresponding Standard Model group action is applied to the triality triple (Ψ+,Ψ−,V) for Ψ+,Ψ−,V∈C⊗H⊗O. Together, Ψ+ and Ψ− provide the correct irreducible representations for two generations. Owing to a certain Cartan factorization, which we define, V provides the irreducible representations for a third generation. Said more explicitly in another way, division algebraic multiplication merges a third generation of spinor representations into a set of scalar bosons. This set of scalar bosons includes the familiar Standard Model Higgs representation.