  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.
