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 $\Theta(x)$ and a spectral function $F(x)$ is minimized over a spectral set $\Omega$, any local minimizer $a$ operator commutes with the Fr\'{ e}chet derivative $\Theta^{ \prime}(a)$. 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.
Categories: math.RA math.OC