Origin of a peculiar property of four particles in a $j=9\mathrm{/}2$ shell

It was found previously that in the system of four particles of the same kind in an angular momentum $j=9\mathrm{/}2$ shell, the energies $E_I$ of four particular levels with angular momenta $I=4$, 6, 10, and 12, are related by $E_I=a+bI(I+1)$, where $a$ and $b$ depend only on the two-body interaction. Here, I discuss observable implications of this regularity and analyse in detail its origin.