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 Cℓ(6). It was previously shown that the minimal left ideals of Cℓ(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.