In this paper we introduce a general notion of a symmetric
cone, valid for the finite and infinite dimensional case,
and prove that one can deduce the seminegative curvature of
the Thompson part metric in this general setting, along with
standard inequalities familiar from operator theory. As a
special case, we prove that every symmetric cone from a JB-algebra
satisfies a certain convexity property for the Thompson part
metric: the distance function between points evolving in time
on two geodesics is a convex function. This provides an affirmative
answer to a question of Neeb [22].