  In this paper, we prove Matsushima's theorem for K\"ahler-Einstein metrics on
a Fano manifold with cone singularities along a smooth divisor that is not
necessarily proportional to the anti-canonical class. We then give an
alternative proof of uniqueness of K\"ahler-Einstein cone metrics by the
continuity method. Moreover, our method provides an existence theorem of
K\"ahler-Einstein cone metrics with respect to conic Ding functional.
