Let $S$ and $T$ be Dirac-type operators on vector bundles $E\rightarrow M$ and $F\rightarrow N$ respectively, where $M$ and $N$ are manifolds. Suppose $Q_S$ and $Q_T$ are parametrices for $S$ and $T$. Form the operator $S\#T$ on the external product bundle $E\boxtimes F\rightarrow M\times N$, defined as $$S\#T=S\otimes 1+1\otimes T.$$

Then $S\#T$ is an elliptic operator and has a parametrix $Q_{S\#T}$ so that

$$1-Q_S S,\qquad 1-Q_T T,\qquad 1-Q_{S\#T}(S\#T),$$ $$1-S Q_S,\qquad 1-T Q_T,\qquad 1-(S\#T)Q_{S\#T}$$ have smooth Schwartz kernels.

Question: How can one construct $Q_{S\#T}$ using $Q_S$ and $Q_T$ so that one can relate the Schwartz kernel of, say $1-Q_{S\#T}(S\#T)$, to those of $1-Q_S S$ and $1-Q_T T$?

Thanks!