  It is set manifest an underlying algebraic structure of Dirac equation and
solutions, in terms of Cl2 Clifford algebra projectors and ladder operators.
From it, a scheme is proposed for constructing unified field theories by
enlarging the pointed algebra. A toy unified matter field model is formulated,
modifying Dirac equation with complex quaternions and octonions. The result
describes a set of fermion fields with reminiscent properties of one standard
model particle generation, exhibiting U(1) electromagnetism, SU(2) flavor and
SU(3) color like symmetries, remarkably SU(2) induces frame fields, though
further explorations are needed.
