Automorphisms of a Clifford-like parallelism

We focus on the description of the automorphism group Gamma(parallel to) of a Clifford-like parallelism parallel to on a 3-dimensional projective double space (P(H-F), parallel to e, parallel to r) over a quaternion skew field H (of any characteristic). We compare Gamma(parallel to) with the automorphism group Gamma(e) of the left parallelism parallel to(e), which is strictly related to Aut(H). We build up and discuss several examples showing that over certain quaternion skew fields it is possible to choose parallel to in such a way that Gamma(parallel to) is either properly contained in Gamma(l) or coincides with Gamma(l) even though parallel to not equal parallel to(l).