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We show that three generations of leptons and quarks with unbroken StandardModel gauge symmetry $SU(3)_c\times U(1)_{em}$ can be described using thealgebra of complexified sedenions $\mathbb{C}\otimes\mathbb{S}$. A primitiveidempotent is constructed by selecting a special direction, and the action ofthis projector on the basis of $\mathbb{C}\otimes\mathbb{S}$ can be used touniquely split the algebra into three complex octonion subalgebras$\mathbb{C}\otimes \mathbb{O}$. These subalgebras all share a commonquaternionic subalgebra. The left adjoint actions of the 8$\mathbb{C}$-dimensional $\mathbb{C}\otimes \mathbb{O}$ subalgebras onthemselves generates three copies of the Clifford algebra $C\ell(6)$. It waspreviously shown that the minimal left ideals of $C\ell(6)$ describe a singlegeneration of fermions with unbroken $SU(3)_c\times U(1)_{em}$ gauge symmetry.Extending this construction from $\mathbb{C}\otimes\mathbb{O}$ to$\mathbb{C}\otimes\mathbb{S}$ naturally leads to a description of exactly threegenerations.

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