Isoparametric hypersurfaces in product spaces of space forms

We give a complete classification of isoparametric hypersurfaces in a product space $M_{{\kappa }_1}^2\times M_{{\kappa }_2}^2$ of 2-dimensional space forms for ${\kappa }_i\in \{-1,0,1\}$ with ${\kappa }_1\ne {\kappa }_2$. In fact we prove that any isoparametric hypersurface in such a space has constant product angle function, which enables us to remove the condition of constant principal curvatures from the classification obtained recently by J.B.M. dos Santos and J.P. dos Santos.