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