Is the diagonal case a general picture for loop quantum cosmology?

The correct implementation of the loop quantum gravity to the early homogeneous Universe has been the subject of a long debate in the literature because the $SU(2)$ symmetry cannot be properly retained. The role of this symmetry is expressed by the Gauss constraint. Here, a nonvanishing Gauss constraint is found. However, we show that using suitable variables, it can be recast into three Abelian constraints, justifying the absence of such a symmetry in loop quantum cosmology.