Geometric commutation principle for weakly spectral sets in Euclidean Jordan algebras

A geometric commutation principle in Euclidean Jordan algebra, recently proved by Gowda, says that, for any spectral set $E$ in a Euclidean Jordan algebra $V$ and $a \in E$, $a$ strongly operator commutes with every element in the normal cone $N_E(a)$. Further, it can be used to establish strong operator commutativity relations in certain optimization problems. Knowing that every spectral sets are special cases of broader class of weakly spectral sets, we prove an analog of a geometric commutation principle for weakly spectral sets and study its consequences and applications.