Three Generations and a Trio of Trialities

We identify the Standard Model’s $\mathfrak{s}\mathfrak{u}(3)\oplus \mathfrak{s}\mathfrak{u}(2)\oplus \mathfrak{u}(1)$ internal symmetries within the triality symmetries $\mathfrak{t}\mathfrak{r}\mathfrak{i}(\mathbb{C})\oplus \mathfrak{t}\mathfrak{r}\mathfrak{i}(\mathbb{H})\oplus \mathfrak{t}\mathfrak{r}\mathfrak{i}(\mathbb{O})$. From here, the corresponding Standard Model group action is applied to the triality triple $({\mathrm{\Psi }}_{+},{\mathrm{\Psi }}_{-},V)$ for ${\mathrm{\Psi }}_{+},{\mathrm{\Psi }}_{-},V\in \mathbb{C}\otimes \mathbb{H}\otimes \mathbb{O}$. Together, ${\mathrm{\Psi }}_{+}$ and ${\mathrm{\Psi }}_{-}$ 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.