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 $S$ 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 $S$ 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 $C\ell$(6), generated from a subset of all left actions of the complex sedenions on themselves. Subsequently we employ the S${}_3$ automorphism of order three, which is an automorphism of $S$ but not of $O$, to generate two additional generations. Given the relative obscurity of sedenions, efforts have been made to present the material in a self-contained manner.