Three fermion generations with two unbroken gauge symmetries from the complex sedenions

We show that three generations of leptons and quarks with unbroken Standard Model gauge symmetry SU(3)${}_c$×U(1)${}_{em}$ can be described using the algebra of complexified sedenions $C$⊗$S$. A primitive idempotent is constructed by selecting a special direction, and the action of this projector on the basis of $C$⊗$S$ can be used to uniquely split the algebra into three complex octonion subalgebras $C$⊗$O$. These subalgebras all share a common quaternionic subalgebra. The left adjoint actions of the 8 $C$-dimensional $C$⊗$O$ subalgebras on themselves generates three copies of the Clifford algebra $C$ℓ(6). It was previously shown that the minimal left ideals of $C$ℓ(6) describe a single generation of fermions with unbroken SU(3)${}_c$×U(1)${}_{em}$ gauge symmetry. Extending this construction from $C$⊗$O$ to $C$⊗$S$ naturally leads to a description of exactly three generations.