  Metric spinor phase of the Infeld-van der Waerden {\gamma}-formalism and
axion field were identified in Ref. [1], by using Maxwell theory. Since axion
couples with fermions, we will investigate Dirac theory to extend the work
given in [1], showing that is possible again to identify this phase with the
axion. By searching an exact identification, we will work yet the spin
transformations to adapt chiral rotations and so to show that the metric spinor
phase behaves exactly as required for the axion field. We will study also the
Dirac-Maxwell system which provides a minimal Lagrangian approach for the axion
classical sector and, finally, we will obtain an explicit 2-spinor description
of magnetic monopole and its charge.
