  Within the framework of the Bhabha-Madhavarao formalism, a consistent
approach to the derivation of a system of the fourth order wave equations for
the description of a spin-$\frac{3}{2}$ particle is suggested. For this purpose
an additional algebraic object, the so-called $q$-commutator ($q$ is a
primitive fourth root of unity) and a new set of matrices $\eta_{\mu}$, instead
of the original matrices $\beta_{\mu}$ of the Bhabha-Madhavarao algebra, are
introduced. It is shown that in terms of the $\eta_{\mu}$ matrices we have
succeeded in reducing a procedure of the construction of fourth root of the
fourth order wave operator to a few simple algebraic transformations and to
some operation of the passage to the limit $z \rightarrow q$, where $z$ is some
(complex) deformation parameter entering into the definition of the
$\eta$-matrices. In addition, a set of the matrices ${\cal P}_{1/2}$ and ${\cal
P}_{3/2}^{(\pm)}(q)$ possessing the properties of projectors is introduced.
These operators project the matrices $\eta_{\mu}$ onto the spins 1/2- and
3/2-sectors in the theory under consideration. A corresponding generalization
of the obtained results to the case of the interaction with an external
electromagnetic field introduced through the minimal coupling scheme is carried
out. The application to the problem of construction of the path integral
representation in parasuperspace for the propagator of a massive
spin-$\frac{3}{2}$ particle in a background gauge field within the
Bhabha-Madhavarao approach is discussed.
