K})$ we can write\
$$\begin{aligned}
\label{2ndcase1}
{\cal O}_{{\cal H}\otimes{\cal H}_{\text{aux}}}\left({\bf K}\right)
 \,=\,
 \prod_{i\in I}{\cal O}^{\text{system}}_{{\cal H}_{I}
  \otimes{\left({\cal H}_{\text{aux}}\right)}_{I}}\left({\bf K}\right)
 \;\,\cdot\!\!\prod_{\substack{\left\{i,i-1,\dots\right\}\\[0.1cm] \subset
       \left\{I,I-1,\dots\right\}}}
  \!{\cal O}^{\text{correlation}}_{{\cal H}_{\left\{i,j,\dots\right\}}
    \otimes{\left({\cal H}_{\text{aux}}\right)}_{\left\{i,j,\dots\right\}}}
     \left({\bf K}\right)\,.\end{aligned}$$ In analogy to the case of the sum decomposition (\[decom1\]) we can apply the RG transformation (\[transf4\]) to (\[2ndcase1\]) which leads to the expression $$\begin{aligned}
\label{2ndcase2}
&{\cal O}_{{\cal H}'\otimes{\cal H}'_{\text{aux}}}\!\!\left({\bf K}'\right)
 \nonumber\\[-0.1cm] & \quad =
 G^+_{{\cal H}\otimes{\cal H}_{\text{aux}}}\circ\left[
 \prod_{i\in I}{\cal O}^{\text{system}}_{{\cal H}_{I}
  \otimes{\left({\cal H}_{\text{aux}}\right)}_{I}}\left({\bf K}\right)
 \cdot\!\!\!\!\prod_{\substack{\left\{i,i-1,\dots\right\}\\[0.1cm] \subset
       \left\{I,I-1,\dots\right\}}}
  \!\!\!\!\!\!{\cal O}^{\text{correlation}}_{{\cal H}_{\left\{i,j,\dots\right\}}
    \otimes{\left({\cal H}_{\text{aux}}\right)}_{\left\{i,j,\dots\right\}}}
     \left({\bf K}\right)\right]
  \circ G_{{\cal H}'\otimes{\cal H}'_{\text{aux}}}\;.\end{aligned}$$ Since this is already the final step in the calculation for this special case of a decomposition we are not able to write the result as a composition of the renormalized system block part and correlation block part as we did in (\[decom5\]) for the sum decomposition. By the considerations so far the product decomposition therefore seems to be not as useful as the sum decomposition for later applications. This is not the case as we will show in the following.\
For the auxiliary space we distinguish between two different cases, an [*active*]{} role and a [*passive*]{} role. Here active means that the auxiliary space is directly involved into the RGT, i.e. 