Characterization of beta distribution on symmetric cones

In the paper we generalize the following characterization of beta distribution to the symmetric cone setting: let XX and YY be independent, non-degenerate random variables with values in (0,1)(0,1), then U=1−XYU=1−XY and V=1−XU are independent if and only if there exist positive numbers pipi, i=1,2,3i=1,2,3, such that XX and YY follow beta distributions with parameters (p1+p3,p2)(p1+p3,p2) and (p3,p1)(p3,p1), respectively.