I am interested in the BRST quantization of the Hilbert-Palatini gravity theory. In the paper https://arxiv.org/abs/gr-qc/9806001, Alexandrov and Vassilevich write down the BRST procedure for defining the path integral of this gravity theory. I am specifically interested in equation (38) and (41) of this paper. These equations are respectively the definition of the gauge fixing fermion they use, and the Lagrangian of the theory with ghosts, in its Hamiltonian form. First, the gauge fixing fermion is: \begin{equation} \psi = -\bar{b}_\alpha n^\alpha+i\bar{c}_\alpha \gamma^{-1}(f^\alpha(q,p)+g^\alpha(n)) \tag{38} \end{equation} Where $\bar{b}$ and $\bar{c}$ are the conjugated ghosts ($b$ is imaginary while $c$ is real), $n^\alpha$ are the Lagrange multipliers in front of the constraints in the action, $\gamma$ is a free parameter, and $f^\alpha$ and $g^\alpha$ are functions we are free to choose. Then they say that if $g^{\alpha}$ do not depend on certain $n^\beta$, and if $f^\alpha$ do not depend on the canonical coordinate $q$, then Equation (41) simplifies and takes the form of a Yang-Mills Lagrangian density: \begin{align} L'_\text{eff} = \dot{q}{}^s p_s+n^\alpha \Phi_\alpha-i\bar{c}_\beta \left( \frac{\partial g^\beta}{\partial n^\alpha}\partial_t-\frac{\partial g^\beta}{\partial n^\gamma}C^\gamma_{\alpha \lambda}n^\lambda + \{\Phi_\alpha,f^\beta\} \right) c^\alpha \tag{41 simplified} \end{align} Where $\Phi_\alpha$ are the constraints (Hamiltonian, diffeomorphism, Gauss and Lorentz), and $C^\gamma_{\alpha \beta}$ are such that $\{\Phi_\alpha,\Phi_\beta\}=C^\gamma_{\alpha \beta} \Phi_\gamma$ (For $\{-,-\}$ the extended Poisson bracket).
I simply would like to know if it was possible, since I believe (but am not sure) we are free to choose $\psi$, to choose $g^i = n^i$ and $f^\alpha=0$, for $i$ belonging to a set such that $n^i$ are precisely the ones simplifying the action?
