Let $B$ be a quaternion algebra over a number field $k$. To a pair of Hilbert symbols $\{a,b\}$ and $\{c,d\}$ for $B$ we associate an invariant $\rho ={\rho }_R\left(\left[𝒟\right(a,b\left)\right],\left[𝒟\right(c,d\left)\right]\right)$ in a quotient of the narrow ideal class group of $k$. This invariant arises from the study of finite subgroups of maximal arithmetic kleinian groups. It measures the distance between orders $𝒟(a,b)$ and $𝒟(c,d)$ in $B$ associated to $\{a,b\}$ and $\{c,d\}.$ If $a=c$, we compute ${\rho }_R(\left[𝒟(a,b)\right],\left[𝒟(c,d)\right])$ by means of arithmetic in the field $k\left(\sqrt{a}\right).$ The problem of extending this algorithm to the general case leads to studying a finite graph associated...