Guided by the Hopf fibration, a family (indexed by a positive constant $K$ ) of right invariant Riemannian metrics on the Lie group $S^3$ is singled out. Using the Yasuda–Shimada paper as an inspiration, a privileged right invariant Killing field of constant length is determined for each $K > 1$ . Each such Riemannian metric couples with the corresponding Killing field to produce a $y$ -global and explicit Randers metric on $S^3$ . Employing the machinery of spray curvature and Berwald's formula, it is proved directly that the said Randers metric has constant positive flag curvature $K$ , as predicted by Yasuda–Shimada. It is explained why this family of Finslerian space forms is not projectively flat.