demonstrated that the three generations of leptons and quarks that are associated with unbroken gauge symmetry \mathrm {SU(3)_{c} \times U(1)_{em}} can be represented using the algebra of the complexified sedenions \mathbb {C \otimes S}. Their reasoning follows that a primitive idempotent projector \rho_{+} = 1/2(1+ie_{15}) — where e_{15} is chosen as an imaginary unit akin to e_{7} for \mathbb {O} in the Fano plane — that acts on the standard basis of the sedenions uniquely divides the algebra into three sets of split basis elements for \mathbb {C \otimes O}, whose adjoint left actions on themselves generate three copies of the Clifford algebra \mathrm Cl(6) which in-turn contain minimal left ideals that describe a single generation of fermions with unbroken \mathrm {SU(3)_{c} \times U(1)_{em}} gauge symmetry. In particular, they note that tensor products between normed division algebras generate zero divisors akin to those inside \mathbb {S}, where for \mathbb {C \otimes O} the lack of alternativity and associativity does not affect the construction of minimal left ideals since their underlying split basis requires only two basis elements to be multiplied together, in-which associativity or alternativity are uninvolved. Still, these ideals constructed from an adjoint algebra of left actions of the algebra on itself remain associative, alternative, and isomorphic to a Clifford algebra. Altogether, this permits three copies of (\mathbb {C \otimes O})_{L} \cong \mathrm {Cl(6)} to exist inside \mathbb {(C \otimes S)}_{L}. Furthermore, these three complexified octonion subalgebras are not independent; they share a common \mathrm Cl(2) subalgebra, which the authors note could form a theoretical basis for CKM and PMNS matrices that, respectively, describe quark mixing and neutrino oscillations.
