$G$ and $G^+$ act nontrivial on this additional space. The commutative diagram describing the general active situation is given in (\[comm2\]).\
$$\label{comm2}
\begin{CD}
{\cal H}'\otimes{\cal H}'_{\text{aux}}
 @> G_{{\cal H}'\otimes{\cal H}'_{\text{aux}}} >> {\cal H}\otimes{\cal H}_{\text{aux}}\\
@V{{\cal O}_{{\cal H}'\otimes{\cal H}'_{\text{aux}}}\left({\bf K}'\right)}
 VV  @VV{{\cal O}_{{\cal H}\otimes{\cal H}_{\text{aux}}}\left({\bf K}\right)}V\\
{\cal H}'\otimes{\cal H}'_{\text{aux}}
 @<<G^+_{{\cal H}\otimes{\cal H}_{\text{aux}}} < {\cal H}\otimes{\cal H}_{\text{aux}}
\end{CD}$$ Relation (\[comm2\]) reduces to a rewriting of (\[comm1\]), if the transformation maps $G$ and $G^+$ each operate as the identity on the auxiliary space and the functional dependence ${\cal O}\left({\bf K}\right)$ acts non trivial only on ${\cal H}$. This gives us an example of the particular case of a passive role of the auxiliary space as it is depicted in (\[comm3\]).\
$$\label{comm3}
\begin{CD}
{\cal H}'\otimes{\cal H}'_{\text{aux}}
 @> G_{{\cal H}'}\otimes{\mathbbm{1}}_{{\cal H}'_{\text{aux}}} >> {\cal H}\otimes{\cal H}_{\text{aux}}\\
@V{{\cal O}_{{\cal H}'\otimes{\cal H}'_{\text{aux}}}\left({\bf K}'\right)}
 VV  @VV{{\cal O}_{{\cal H}\otimes{\cal H}_{\text{aux}}}\left({\bf K}\right)}V\\
{\cal H}'\otimes{\cal H}_{\text{aux}}
 @<<G^+_{\cal H}\otimes{\mathbbm{1}}_{{\cal H}_{\text{aux}}} < {\cal H}\otimes{\cal H}_{\text{aux}}
\end{CD}$$ In the case of (\[comm2\]) we can think of the auxiliary space as some kind of medium not changed during a RG step. The active and the passive choice of the auxiliary space yield two different realizations of our RG, which we will call the ’general (real-space) RG’ 