Hilbert space and commutation relations are in both cases the ones for the $SU(2)$ group. The distinction from 3D to 4D arises only in the dynamics, with the 4D gauge symmetry $SL(2,C)$ or $SU(2)\times SU(2)$ implemented on the bulk of the spin foam.

The dynamics are implemented in the form of a state sum model with quantum transition amplitudes or partition functions. Let us consider first the transition amplitude. It is a function of the states defined on $\Gamma=(\partial\triangle)^*$, the boundary graph. This can be defined in the group representation $W_{\triangle}(U_l)$ or in the so called spin representation $W_{\triangle}(j_f)$, where the $\triangle$ indicates that the amplitude is computed in a discretization $\triangle$ of the bulk that matches the boundary graph. In $W_{\triangle}(j_f)$, a basis on the gauge invariant Hilbert space is given by the normalized eigenvectors of the operator $L_l$, indicated by $\ket{j_f}$. An element of this basis is determined by assigning a spin $j_l$ of a representation of $SU(2)$ to each link $l$ of the graph. A graph with a spin assigned to each link is called a spin network. So the spin network states $\ket{j_f}$ form a basis of the gauge invariant Hilbert space of quantum gravity, i.e. 