mapping from the dense domain in $\Hil_{\gr}$ of the Bianchi I model to the one the isotropic model as follows $$\psi(\lambda_1,\lambda_2,v) \mapsto \psi(v) = [{\hat{\boldsymbol{P}}}\psi](v) 
  := \sum_{\lambda_1,\lambda_2}\psi(\lambda_1,\lambda_2,v) .$$ Through the direct inspection one can check that there exists an operator $\bar{\Theta}_{\operatorname*{B1}}$ such that $$[{\hat{\boldsymbol{P}}}\Theta_{\operatorname*{B1}}\psi](v) = \bar{\Theta}_{\operatorname*{B1}} [{\hat{\boldsymbol{P}}}\psi](v) .$$ An action of that operator equals exactly the one of $\Theta_{\operatorname*{sLQC}}$ defined via (\[eq:ev2-gen\], \[eq:fslqc\]).

The consequence of the above observation is, that at least to some extent those of the aspects of the Bianchi I model, which are related exclusively to the behavior of the isotropic degrees of freedom, can be investigated via an isotropic model constructed via averaging over anisotropies and equivalent to the sLQC one described in Sect. \[sec:frame-FRW\]. However, one has to be aware, that some of the physical states might in principle be in the kernel of the projection operator ${\hat{\boldsymbol{P}}}$. Thus, certain care needs to be taken, when relating the properties of the isotropic sector defined above with the full Bianchi I model. In particular it is not confirmed, whether the expectation values and dispersions of the total volume Bianchi I operators agree with the analogous quantities of the volume operators acting on the averaged states. 