G_{{\cal H}'_{I}\,\otimes\,
        {\left({\cal H}'_{\text{aux}}\right)}_{I}}\right]\nonumber\\[0.2cm]
    &\qquad\qquad\qquad\qquad\cdot\prod_{\substack{J\in {\frak I}
        \\[0.1cm] J\not= I}}
  G^+_{{\cal H}_{J}\,\otimes\, {\left({\cal H}_{\text{aux}}\right)}_{J}}
  \,\circ\,
  G_{{\cal H}'_{J}\,\otimes\, {\left({\cal H}'_{\text{aux}}\right)}_{J}}\;.\end{aligned}$$ This is exactly the local RGT for the system blocks (\[grad3\]) if we neglect the last product term on the right hand side of (\[decom3\]). We will refer to this factor as a correction term that vanishes if we demand $$\begin{aligned}
\label{decom4}
 G^+_{{\cal H}\otimes{\cal H}_{\text{aux}}}\,\circ\,
   G_{{\cal H}'\otimes{\cal H}'_{\text{aux}}}
   \,=\,{\mathbbm{1}}_{{\cal H}'\otimes{\cal H}'_{\text{aux}}}\;.\end{aligned}$$ Inserting this constraint into (\[decom3\]), carrying out the same calculation for the correlation blocks and finally using relation (\[grad3\]) we get the renormalized version of equation (\[decom1\]) given by $$\begin{aligned}
\label{decom5}
{\cal O}_{{\cal H}'\otimes{\cal H}'_{\text{aux}}}\left({\bf K}'\right)
 \,=\,
 \sum_{I\in {\frak I}}{\cal O}^{\text{system}}_{{\cal H}'_{I}
  \otimes{\left({\cal H}'_{\text{aux}}\right)}_{I}}\left({\bf K}'\right)
 \,+\sum_{\substack{\left\{i,i-1,\dots\right\}\\[0.1cm] \subset
       \left\{I,I-1,\dots\right\}}}
  \!{\cal O}^{\text{correlation}}_{{\cal H}'_{\left\{i,i-1,\dots\right\}}
    \otimes{\left({\cal H}'_{\text{aux}}\right)}_{\left\{i,i-1,\dots\right\}}}
     \left({\bf K}'\right)\,,\end{aligned}$$ which leads to the renormalized set of parameters ${\bf K}'$. In (\[decom5\]) we used the reasonable definition $$\label{grad6}
\begin{split}
{\cal O}^{\text{correlation}}_{{\cal H}'_{\left\{i,i-1,\dots\right\}}
  \otimes{\left({\cal H}'_{\text{aux}}\right)}_{\left\{i,i-1,\dots\right\}}}
  ({\bf K}')
\,:=\,& \left[G^+_{{\cal H}_{\left\{i,i-1,\dots\right\}}\,\otimes\,
    {\left({\cal H}_{\text{aux}}\right)}_{\left\{i,i-1,\dots\right\}}}
\right]\\[0.3cm]
   &\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\circ\,
    {{\cal O}^{\text{correlation}}_{{\cal H}_{\left\{i,i-1,\dots\right\}}\otimes
  {\left({\cal H}_{\text{aux}}\right)}_{\left\{i,i-1,\dots\right\}}}
     \!\left({\bf K}\right)}
      \,\circ\,\left[G_{{\cal H}'_{\left\{i,i-1,\dots\right\}}\,\otimes\,
    {\left({\cal H}'_{\text{aux}}\right)}_{\left\{i,i-1,\dots\right\}}}\right]
  \;.\\[0.2cm]
\end{split}$$ Relation (\[decom4\]) introduces an additional constraint for the RGT and therefore restricts the variety of possible transformations.\
In the case of a product decomposition of the operator ${\cal O}({\bf 