  We present a necessary and sufficient condition for the strict positive
definiteness of a real, continuous, isotropic and positive definite kernel on a
two-point compact homogeneous space. The characterization adds to others
previously obtained by D. Chen at all (2003) in the case in which the space is
a sphere of dimension at least 2 and Menegatto at all (2006) in the case in
which the space is the unit circle. As an application, we use the
characterization to improve upon a recent result on the differentiability of
positive definite kernels on the spaces.
