  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.
