Commutation principles in Euclidean Jordan algebras and normal decomposition systems

The commutation principle of Ramirez, Seeger, and Sossa \cite{ramirez-seeger-sossa} proved in the setting of Euclidean Jordan algebras says that when the sum of a Fr'{e}chet differentiable function $\mathrm{\Theta }(x)$ and a spectral function $F(x)$ is minimized over a spectral set $\mathrm{\Omega }$, any local minimizer $a$ operator commutes with the Fr'{e}chet derivative ${\mathrm{\Theta }}^{\mathrm{\prime }}(a)$. 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.