Hyperbolicity cones, and in particular symmetric cones, are of great interest in optimization. Renegar showed that every hyperbolicity cone has a family of derivative cones that approximate it. Ito and Lourenço found the automorphisms of those derivatives when the original cone is generated by rank-one elements, as symmetric cones happen to be. We show that the derivative automorphisms of a symmetric cone are closely related to the automorphisms of its associated Euclidean Jordan algebra. In the process, we find the automorphism group of the quaternion positive-semidefinite cone and list explicitly the Jordan-automorphisms of the quaternion Hermitian matrices. We also address the path-connectedness of the simple Euclidean Jordan-automorphism groups.
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