  A Fan-Theobald-von Neumann system is a triple $(V,W,\lambda)$, where $V$ and
$W$ are real inner product spaces and $\lambda:V \to W$ is a norm-preserving
map satisfying a Fan-Theobald-von Neumann type inequality together with a
condition for equality. Examples include Euclidean Jordan algebras, systems
induced by certain hyperbolic polynomials, and normal decompositions systems
(Eaton triples). In the previous paper (arXiv:1902.06640) we presented some
basic properties of such systems and described results on optimization problems
dealing with certain combinations of linear/distance and spectral functions. We
also introduced the concept of commutativity via the equality in the
Fan-Theobald-von Neumann type inequality. In the present paper, we elaborate on
the concept of commutativity and introduce/study automorphisms, majorization,
and reduction in Fan-Theobald-von Neumann systems.
