  We show that three generations of leptons and quarks with unbroken Standard
Model gauge symmetry $SU(3)_c\times U(1)_{em}$ 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 $C\ell(6)$. It was
previously shown that the minimal left ideals of $C\ell(6)$ describe a single
generation 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 three
generations.
