Generators of the exceptional group E8 as bilinear quark and lepton fields

The quarks and leptons are assigned to the adjoint representation of the exceptional group E8 using decompositions under the subgroups SU(9) and [SU(3)]4. Generators are constructed as linear combinations of bilinear quark and lepton fields. Closure of the algebra is used to determine the unknown coefficients of the linear combinations. It is noted that the Majorana spinors introduced to represent the adjoint representations of SU(9) and [SU(3)]4 subgroups cannot be taken traceless. The trace should couple to the quark and lepton fields in order to close the algebra. The constraints on the bilinear fields which are of physical importance are introduced to obtain the right number of fermionic states in the adjoint representation. An attractive possibility of having an octet of strictly massless Majorana quarks and at least three massless Majorana leptons as a consequence of pure algebraic constraints is discussed. The exceptional subgroups E7 and E6 are identified and the explicit commutation relations are obtained. Using one assignment of E6 the role of color-singlet lepton-lepton and quark-antiquark currents is pointed out.