Strictly positive definite kernels on a product of spheres

For the real, continuous, isotropic and positive definite kernels on a product of spheres, one may consider not only its usual strict positive definiteness but also strict positive definiteness restrict to the points of the product that have distinct components. In this paper, we provide a characterization for strict positive definiteness in these two cases, settling all the cases but those in which one of the spheres is a circle.