  A generalization of the Lagrangian introduced earlier in [2011 {\it J. Phys.
G} ${\bf 37}$ 105001] for a classical color spinning particle interacting with
background non-Abelian gauge and fermion fields for purpose of considering a
change in time of the spin particle degree of freedom, is suggested. In the
case under consideration the spin degree of freedom is described by a commuting
$c$-number Dirac spinor $\psi_{\alpha}$. A mapping of this spinor into new
variables: anticommuting pseudovector $\xi_{\mu}$ and pseudoscalar $\xi_5$
commonly used in a description of the spin degree of freedom of a massive
spin-1/2 particle, is constructed. An analysis of one-to-one correspondence of
this mapping is given. It is shown that for the one-to-one correspondence it is
necessary to extend a class of real tensor quantities including besides
$\xi_{\mu}$ and $\xi_5$, also odd vector $\hat{\xi}_{\mu}$, scalar
$\hat{\xi}_5$, and (dual) pseudotensor $^{\ast}\zeta_{\mu \nu}$. In addition,
it is shown that it is necessary either to restrict a class of the initial
spinor $\psi_{\alpha}$ to Majorana one or to double the number of variables in
the tensor aggregate $(\xi_{\mu},\,\xi_5,, ^{\ast}\zeta_{\mu
\nu},\,\hat{\xi}_{\mu},\,\hat{\xi}_5)$. Various special cases of the desired
mapping are considered. In particular, a connection with the Lagrangian
suggested by A.M. Polyakov, is studied. It is also offered the way of obtaining
the supersymmetric Lagrangian in terms of the even $\psi_{\alpha}$ and odd
$\theta_{\alpha}$ spinors. The map of the Lagrandian leads to local SUSY
Lagrangian in terms of $\xi_{\mu}$ and $\xi_5$.
