\sqrt{{{\,{}^o\! q}}} {{\,{}^o\! e}}^a_i , 
\tag{\ref{eq:Ashtekar-B1}}\end{aligned}$$

and all the constraints except the Hamiltonian one are automatically solved. The Poisson brackets between coefficients equal $\{c^i,p_j\}=8\pi G\gamma\delta^i_j$.

Following [@awe] we choose the lapse $N=\sqrt{|p_1p_2p_3|}$. The Hamiltonian constraint has the same form as in the FRW case, and its components are given by and (with the basic variables given by instead of ), where in $p:=|p_1p_2p_3|^{1/3}$.

### Loop quantization {#loop-quantization}

To quantize the system we follow the program specified in Sect. \[sec:frame-FRW-q\], in particular choosing the polymer representation for the geometry degrees of freedom, while keeping the Schroedinger one for the scalar field.

In the geometry part the basic operators are holonomies along straight edges generated by ${{\,{}^o\! e}}^a_i$ and the fluxes across $2$-dimensional rectangles spanned by ${{\,{}^o\! e}}^a_i$. The gravitational kinematical Hilbert space consists of the product of three copies of $\Hil_{\gr}$ of the FRW system, each corresponding to one direction of ${{\,{}^o\! e}}^a_i$ $\Hil_{\gr} = \bigotimes_{i=1}^3 L^2(\bar{\re},\rd\mu_{\Bohr}) .$ The basis of this space can be built out of eigenstates of the triad (or unit flux) operators $\hat{p}_i$ and parametrized by three real variables $\lambda_i$ such that $\hat{p}_i\ket{\lambda_1,\lambda_2,\lambda_3}
=\sgn(\lambda_i)(4\pi\gamma\sqrt{\Delta}\lPl^3)^{\frac{2}{3}}\, \lambda_i^2\,
\ket{\lambda_1,\lambda_2,\lambda_3}$. Alternatively, one of $\lambda_i$ can be replaced with the parameter $v:=2\lambda_1\lambda_2\lambda_3$. 