Three generations of colored fermions with $S_3$ family symmetry from Cayley-Dickson sedenions

An algebraic representation of three generations of fermions with $SU(3{)}_C$ color symmetry based on the Cayley-Dickson algebra of sedenions $\mathbb{S}$ is constructed. Recent constructions based on division algebras convincingly describe a single generation of leptons and quarks with Standard Model gauge symmetries. Nonetheless, an algebraic origin for the existence of exactly three generations has proven difficult to substantiate. We motivate $\mathbb{S}$ as a natural algebraic candidate to describe three generations with $SU(3{)}_C$ gauge symmetry. We initially represent one generation of leptons and quarks in terms of two minimal left ideals of $\mathbb{C}\mathrm{\ell }(6)$, generated from a subset of all left actions of the complex sedenions on themselves. Subsequently we employ the finite group $S_3$, which are automorphisms of $\mathbb{S}$ but not of $\mathbb{O}$ to generate two additional generations. Given the relative obscurity of sedenions, efforts have been made to present the material in a self-contained manner.