  It is proved that there exist a vector representation of Dirac's spinor field
and in one sense it is equivalent to biquaternion (i.e. complexified
quaternion) representation. This can be considered as a generalization of
Cartan's idea of triality to Dirac's spinors. In the vector representation the
first order Dirac Lagrangian is dual equivalent to the two order Lagrangian of
topologically massive gauge field. The potential field which corresponds to the
Dirac field is obtained by using master (or parent) action approach. The novel
gauge field is self-dual and contains both anti-symmetric Lie and symmetric
Jordan structure.
