Relation to a property of the angular momentum zero space of states of four fermions in an angular momentum $j=9\mathrm{/}2$ shell unexpectedly found to be stationary for any rotationally invariant two-body interaction

The existence of states with angular momenta $I=4$ and~6 of four fermions in an angular momentum $j=9\mathrm{/}2$ shell that are stationary for any rotationally invariant two-body interaction despite the presence of other states with the same angular momentum, the Escuderos-Zamick states, is shown to be equivalent to the invariance to any such interaction of the span of states generated from $I=0$ states by one-body operators. This invariance is verified by exact calculation independently of previous verifications of the equivalent statement. It explains the occurrence of the Escuderos-Zamick states for just $I=4$ and 6. The action of an arbitrary interaction on the invariant space and its orthogonal complement is analyzed, leading to a relation of the Escuderos-Zamick energy levels to levels with $I=10$ and 12. Parts of the observed spectra of ${}^{72}$Ni, ${}^{74}$Ni, ${}^{94}$Ru, ${}^{96}$Pd, ${}^{212}$Rn, and ${}^{214}$Ra are discussed in the light of this relation.