Three generations, two unbroken gauge symmetries, and one eight-dimensional algebra

A considerable amount of the standard model's three-generation structure can be realised from just the $8\hspace{.3mm}\mathbb{C}$-dimensional algebra of the complex octonions. Indeed, it is a little-known fact that the complex octonions can generate on their own a $64\hspace{.3mm}\mathbb{C}$-dimensional space. Here we identify an $su(3)\oplus u(1)$ action which splits this $64\hspace{.3mm}\mathbb{C}$-dimensional space into complexified generators of $SU(3)$, together with 48 states. These 48 states exhibit the behaviour of exactly three generations of quarks and leptons under the standard model's two unbroken gauge symmetries. This article builds on a previous one, [1], by incorporating electric charge. Finally, we close this discussion by outlining a proposal for how the standard model's full set of states might be identified within the left action maps of $\mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$ (the Clifford algebra $\mathbb{C} l(8)$). Our aim is to include not only the standard model's three generations of quarks and leptons, but also its gauge bosons.