We characterize the Wishart distributions on a symmetric cone C. If $C = (0, +\infty)$, this has been done by Lukacs in 1955. If C is the cone of positive definite symmetric matrices, this has been done by Olkin and Rubin in 1962. We both shorten and extend the Olkin-Rubin proof (sometimes obscure) by using three modern ideas: (i) try to avoid artificial coordinates in differential geometry; (ii) the variance function of a natural exponential family F characterizes F; (iii) symmetric matrices are a particular example of a Euclidean simple Jordan algebra.
"The Lukacs-Olkin-Rubin characterization of Wishart distributions on symmetric cones." Ann. Statist. 24 (2) 763 - 786, April 1996. https://doi.org/10.1214/aos/1032894464