without any assumptions to nonvanishing $\beta$-function [@Mue91]: $$\begin{aligned}
\label{conf-constr-KD-full}
&&\left[\hat{a}(l)+\hat{\gamma}^c(l)
+2{\beta\over g}\hat{b}(l)
,\hat{\gamma}\right]=0,\end{aligned}$$ where the $\hat{b}$ matrix is defined as[^1] $$\begin{aligned}
\label{def-b}
b_{jj'}(l)= \theta_{jj'}
\left\{ 2(l+j'+3)\delta_{jj'} - [1 + (-1)^{ j - j'}] (2j' + 3)
\right\} .\end{aligned}$$ Hence, the contribution to the off-diagonal matrix elements of the anomalous dimension matrix due to $\beta$-function is predicted too. Under the conjecture that only gauge invariant composite operators contribute to the constraints for the gauge independent anomalous dimension matrix of the conformal operators, which turns out to be quite natural, formally the same equation can also be derived in Abelian gauge field theory for the flavour non-singlet channel [@Mue94] as well as for the quark-gluon singlet sector studied in the present paper. The knowledge of the conformal anomaly matrix allows to reconstruct the off-diagonal part of the anomalous dimension matrix for the corresponding channels.

Few remarks are in order at this point. Although the above mentioned commutator constraints are derived, for the reasons of technical simplification, in the theory with $U(1)$ symmetry group, nevertheless, they are valid in NLO approximation for $QQ$- and $QG$-channels we have considered, also in the non-Abelian theory. This is true since in the conformal limit, the symmetry 