Summary: By interpreting J. A. Lester’s [Can. Math. Bull. 34, No. 4, 492–498 (1991; Zbl 0757.51005)] result on inversive-distance-preserving mappings as an axiomatizability statement, and by using the Liebmann isomorphism between the inversive plane and hyperbolic three-space, we point out that hyperbolic three-spaces (and inversive geometry) coordinatized by Euclidean fields can be axiomatized with planes (or circles) as variables, by using only the plane-orthogonality (or circle-orthogonality) predicate \(\perp_p\) (or \(\perp_c)\), or by using only the predicate \(\delta'\) (or \(\delta)\), where \(\delta'(p,p')\) (or \(\delta(A,B))\) is interpreted as ‘the distance between the planes \(p\) and \(p'\) is equal to the length of the segment \(s\) whose angle of parallelism is \(\frac{\pi} {4}\) (i.e. \(\Pi(s)=\frac {\pi}{4})\)’ (or as ‘the numerical distance between the disjoint circles \(A\) and \(B\) has the value \(\rho\) which corresponds to \(s\) via Liebmann’s isomorphism’).