The commutation principle of Ramirez, Seeger, and Sossa proved in the setting of Euclidean Jordan algebras says that when the sum of a real valued function $h$ and a spectral function $Phi$ is minimized/maximized over a spectral set $E$, any local optimizer $a$ at which $h$ is Fr{e}chet differentiable operator commutes with the derivative $h^{prime}(a)$. In this paper, assuming the existence of a subgradient in place the derivative (of $h$), we establish `strong operator commutativity relations: If $a$ solves the problem $underset{E}{max},(h+Phi)$, then $a$ strongly operator commutes with every element in the subdifferential of $h$ at $a$; If $E$ and $h$ are convex and $a$ solves the problem $underset{E}{min},h$, then $a$ strongly operator commutes with the negative of some element in the subdifferential of $h$ at $a$. These results improve known (operator) commutativity relations for linear $h$ and for solutions of variational inequality problems. We establish these results via a geometric commutation principle that is valid not only in Euclidean Jordan algebras, but also in the broader setting of FTvN-systems.
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