Standard Model Symmetries and the Nested Embeddings of $\mathbb{R}\subset\mathbb{C}\subset\mathbb{H}\subset\mathbb{O}$

Where do the Standard Model's internal symmetries come from? Treating $\mathbb{O}\oplus\mathbb{H}\oplus\mathbb{C}\oplus\mathbb{R}$ as a module for its own multiplication algebra enables a particular origin story for the Standard Model's pre-Higgs, $\mathfrak{g}_{SM}:=\mathfrak{su}(3)_{C} \oplus \mathfrak{su}(2)_{L} \oplus \mathfrak{u}(1)_{Y},$ and post-Higgs, $\mathfrak{g}_{LE}:=\mathfrak{su}(3)_{C} \oplus \mathfrak{u}(1)_{Q},$ symmetries. We recognize both these endomorphisms and their modules alike as $\mathbb{Z}_2^n$-graded algebras. Then, annihilating certain highest grade (volume) elements, and imposing an equal-trace condition on anti-hermitian operators leads precisely to $\mathfrak{g}_{SM}$ and $\mathfrak{g}_{LE}$. Weak hypercharge and electric charge operators, $Y$ and $Q,$ take on a remarkably simple form: $\sum \frac{1}{n}\mathbb{I}_{n\times n}$. With the help of auxiliary imaginary units, this 15 $\mathbb{R}$ dimensional $\mathbb{O}\oplus\mathbb{H}\oplus\mathbb{C}\oplus\mathbb{R}$ embeds naturally as a vector space into several well-studied 16 $\mathbb{R}$ dimensional algebras, which we generically refer to as $\mathbb{V}.$ With this embedding, the Standard Model's internal symmetries may then be seen to arise in part from the sequence of nested inclusions: $\mathbb{R}\subset\mathbb{C}\subset\mathbb{H}\subset \mathbb{O}\subset\mathbb{V}.$ In the sedenionic case of $\mathbb{V} = \mathbb{S},$ the full sequence becomes a Cayley-Dickson tower. We define the notion of endomorphic models of particle physics, and connect $End_\mathbb{R}(\mathbb{V})\simeq Cl(0,8)$ to the earlier ideas of Bott Periodic Particle Physics. We comment on a possible connection between the existence of multiple complex structures and the baryon asymmetry problem.