The Lukacs–Olkin–Rubin theorem on symmetric cones through Gleason's theorem
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–Wesołowski, 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.