In this paper we characterize symmetric cones among homogeneous convex cones by the condition that the corresponding tube domains are mapped onto the dual tube domains under pseudoinverse maps with parameter. The condition also restricts the parameter to specific ones.
Chifune KAI. Takaaki NOMURA. "A characterization of symmetric cones through pseudoinverse maps." J. Math. Soc. Japan 57 (1) 195 - 215, January, 2005. https://doi.org/10.2969/jmsj/1160745822