There are three planar hypercomplex number systems, which we shall denote $\mathbb C, \mathbb D, \mathbb R^2$. It is known that the projective lines over these number systems correspond to the Laguerre spears in elliptic, Euclidean and hyperbolic Laguerre geometries respectively. It is known that for each $A \in \{\mathbb C, \mathbb D, \mathbb R^2\}$, the corresponding projective general linear group $PGL_2(A)$ is isomorphic to the corresponding elliptic/Euclidean/hyperbolic Laguerre transformations which don't reverse orientation.
This is known, and there are ways to verify this using (let's say) Lie Sphere Geometry, or by decomposing the elements in $PGL_2(A)$, but what is the easiest way of seeing this? The material I've found so far is in German or Russian, by either Walter Benz or Isaak Yaglom. I'm requesting references.