Gauge fixing and abelianization in simple BRST quantization

In a previous paper \cite{Simple} it was shown that the BRST charge $Q$ for any gauge model with a Lie algebra symmetry may be decomposed as $Q=\del+\del^{\dag},\;\;\;\del^2=\del^{\dag 2}=0,\;\;\;[\del, \del^{\dag}]_+=0$ provided dynamical Lagrange multipliers are used but without introducing other matter variables in $\del$ than the gauge generators in $Q$. In this paper further decompositions are derived but now by means of gauge fixing operators. As in \cite{Simple} it is shown that $\del=c^{\dag a}\phi_a$ where $c^a$ are new ghosts and $\phi_a$ are nonhermitian variables satisfying the gauge algebra. However, in distinction to \cite{Simple} also solutions of the form $\del=c^{\dag a}A_a$ where $A_a$ satisfy an abelian algebra is derived (abelianization). By means of a bigrading the BRST condition reduces to $\del|ph\hb=\del^{\dag}|ph\hb=0$ on inner product spaces whose general solutions are expressed in terms of the solutions to a proper Dirac quantization. Thus, the procedure provides for inner products for the solutions of a Dirac quantization.