  We prove the Lukacs characterization of the Wishart distribution on
non-octonion symmetric cones of rank greater than 2. We weaken the smoothness
assumptions in the version of the Lukacs theorem of [Bobecka-Weso{\l}owski,
Studia Math. 152 (2002), 147-160]. The main tool is a new solution of the
Olkin-Baker functional equation on symmetric cones, under the assumption of
continuity of respective functions. It was possible thanks to the use of
Gleason's theorem.
