Commutation principles in Euclidean Jordan algebras and normal decomposition systems

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The commutation principle of Ramirez, Seeger, and Sossa ramirez-seeger-sossa proved in the setting of Euclidean Jordan algebras says that when the sum of a Fréchet differentiable function Θ(x) and a spectral function F(x) is minimized over a spectral set Ω, any local minimizer a operator commutes with the Fréchet derivative Θ′(a). In this paper, we extend this result to sets and functions which are (just) invariant under algebra automorphisms. We also consider a similar principle in the setting of normal decomposition systems.