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 ρ = ρ R [ 𝒟 ( a , b ) ] , [ 𝒟 ( c , d ) ] 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 ρ R ( [ 𝒟 ( a , b ) ] , [ 𝒟 ( c , d ) ] ) by means of arithmetic in the field k ( a ) . The problem of extending this algorithm to the general case leads to studying a finite graph associated to different Hilbert symbols for B . An example arising from the determination of the smallest arithmetic hyperbolic 3 -manifold is discussed.