An unorthodox unified theory is developed to describe external and internal
attributes and symmetries of fundamental fermions, quarks and leptons. Basic
ingredients of the theory are an algebra which consists of all the
triple-direct-products of Dirac gamma-matrices and a triple-spinor-field,
called a triplet field, defined on the algebra. The algebra possesses three
commutative sub-algebras which describe, respectively, the external space-time
symmetry, the family structure and the internal color symmetry of quarks and
leptons. The triplet field includes threefold (fourfold) repetitionary modes of
spin 1/2 component fields with SU(3) (SU(4)) color symmetry. It is possible to
qualify the Yukawa interaction and to make a new interpretation of its coupling
constants naturally in an intrinsic mechanism of the triplet field formalism.
The Dirac mass matrices with quasi-democratic structure are derived as an
illustration