  We consider (effective) Quantum General Relativity coupled to the Standard
Model (QGR-SM) and study the corresponding BRST double complex. This double
complex is generated by infinitesimal diffeomorphisms and infinitesimal gauge
transformations. To this end, we define the respective BRST differentials $P$
and $Q$ and their corresponding anti-BRST differentials $\overline{P}$ and
$\overline{Q}$. Then we show that all differentials mutually anticommute, which
allows us to introduce the total BRST differential $D := P + Q$ and the total
anti-BRST differential $\overline{D} := \overline{P} + \overline{Q}$.
Furthermore, we characterize all non-constant Lagrange densities that are
essentially closed with respect to the diffeomorphism differentials $P$ and
$\overline{P}$ as scalar tensor densities of weight $w = 1$. As a consequence
thereof, we obtain that graviton-ghosts decouple from matter if the Yang--Mills
theory gauge fixing and ghost Lagrange densities are tensor densities of weight
$w = 1$. In particular, we show that every such Lagrange density can be
modified uniquely to satisfy said condition. Finally, we introduce a total
gauge fixing fermion $\Upsilon$ and show that we can generate the complete
gauge fixing and ghost Lagrange densities of QGR-SM via $D \Upsilon$.
