It is shown how the algebra $ {{\bf J } }_3 [ { \bf C \otimes O } ] \otimes {Cl(4, {\bf C}) } $ based on the tensor product of the complex Exceptional Jordan $ {{\bf J } }_3 [ { \bf C \otimes O } ]$, and the complex Clifford algebra $ Cl(4, {\bf C}) $, can describe all of the spinorial degrees of freedom of three generations of fermions in four-spacetime dimensions, and, in addition, to include the degrees of freedom of three sets of pairs of complex scalar Higgs-doublets $\{ {\bf H}^i_L, {\bf H}^i_R\}; i = 1,2,3$, and their conjugates. A close inspection of the fermion structure of each generation reveals that it fits naturally with the $ {\bf 16}$ complex-dimensional representation of the internal left/right symmetric gauge group $ G_{LR} = SU(3)_C \times SU(2)_L \times SU(2)_R \times U(1)$. It is reviewed how the latter group emerges from the intersection of $ SO (10) $ and $ SU(3) \times SU(3) \times SU(3) $ in $E_6$. In the concluding remarks we briefly discuss the role that the extra Higgs fields may have as dark matter candidates; the construction of Chern-Simons-like matrix cubic actions; hexaquarks and Clifford bundles over the complex-octonionic projective plane $ { \bf (C \otimes O) P^2 } $ whose isometry group is $ E_6$.