  We construct the Rarita-Schwinger basis vectors, $U^\mu$, spanning the direct
product space, $U^\mu:=A^\mu\otimes u_M$, of a massless four-vector, $ A^\mu $,
with massless Majorana spinors, $u_M$, together with the associated
field-strength tensor, ${\mathcal T}^{\mu\nu}:=p^\mu U^\nu -p^\nu U^\mu$. The
${\mathcal T}^{\mu\nu}$ space is reducible and contains one massless subspace
of a pure spin-$3/2$ $\in (3/2,0)\oplus (0,3/2)$. We show how to single out the
latter in a unique way by acting on ${\mathcal T}^{\mu\nu}$ with an earlier
derived momentum independent projector, ${\mathcal P}^{(3/2,0)}$, properly
constructed from one of the Casimir operators of the algebra $so(1,3)$ of the
homogeneous Lorentz group. In this way it becomes possible to describe the
irreducible massless $(3/2,0)\oplus (0,3/2)$ carrier space by means of the
anti-symmetric-tensor of second rank with Majorana spinor components, defined
as $\left[ w^{(3/2,0) }\right]^{\mu\nu}:=\left[{\mathcal
P}^{(3/2,0)}\right]^{\mu \nu}\,\,_{\gamma\delta}{\mathcal T}^{\gamma \delta }$.
The conclusion is that the $(3/2,0)\oplus (0,3/2)$ bi-vector spinor field can
play the same role with respect to a $U^\mu$ gauge field as the bi-vector,
$(1,0)\oplus (0,1)$, associated with the electromagnetic field-strength tensor,
$F_{\mu\nu}$, plays for the Maxwell gauge field, $A_\mu$. Correspondingly, we
find the free electromagnetic field equation, $p^\mu F_{\mu\nu}=0$, is
paralleled by the free massless Rarita-Schwinger field equation, $p^\mu \left[
w^{(3/2,0)}\right]_{\mu\nu}=0$, supplemented by the additional condition,
$\gamma^\mu\gamma^\nu \left[ w^{(3/2,0)}\right]_{\mu \nu} =0$, a constraint
that invokes the Majorana sector.
