within the RGT.\
To find a solution for this problem our Ansatz is to enlarge the Hilbert space ${\cal H}$ by an additional (auxiliary) Hilbert space ${\cal H}_{\text{aux}}$ due to the composition rule $$\label{total}
 {\cal H}_{\text{total}} \,=\, {\cal H}\,\otimes\,{\cal H}_{\text{aux}}\;.$$ We like to think of the space ${\cal H}_{\text{total}}$ as some kind of ’super space’ and the global operator ${\cal O}_{{\cal H}\otimes {\cal H}_{\text{aux}}}\left({\bf K}\right)$ is then embedded into the total space ${\cal H}_{\text{total}}$. The key idea is to recover a block decomposition for ${\cal O}_{{\cal H}\otimes {\cal H}_{\text{aux}}}\left({\bf K}\right)$ into blocked pieces of the form ${{\cal O}_{{\cal H}_{I}\otimes{\left({\cal H}_{\text{aux}}\right)}_{I}}
  \left({\bf K}\right)}$ which we identify as superblocks according to section \[dmrgintro\]. The next step in our approach following the basic principles of DMRG is to outline a general construction for ${\cal O}_{{\cal H}\otimes {\cal H}_{\text{aux}}}\left({\bf K}\right)$ with a commuting block decomposition. This can be performed explicitly by starting with standard real-space RG concepts.\
In the formulation of [*standard*]{} block RG we consider a decomposition of ${\cal O}_{{\cal H}\otimes {\cal H}_{\text{aux}}}\left({\bf K}\right)$ into disconnected block functions given by $$\begin{aligned}
\label{grad4b}
{\cal O}_{{\cal H}_{I}
  \otimes{\left({\cal H}_{\text{aux}}\right)}_{I}}({\bf K})
 \,=\,{\cal O}^{\text{system}}_{{\cal H}_{I}
  \otimes{\left({\cal H}_{\text{aux}}\right)}_{I}}({\bf K})
   \qquad\text{with}\quad I\in {\frak I}\,.\end{aligned}$$ where we have neglected the 