degeneracy between BC/PL and that betweenAFM$^*_{a/b}$.[]{data-label="Fig:PD_wring"}](SFPD_wring_J3_1_K1=K3.eps){width="2"}

*Discussions.*— We close by remarking on several points. The small ring exchanges do not affect dramatically the energetics of $(\pi,0)$ collinear AFM phase and the $(\pi,0)$ antiferro-quadrupolar order (AFQ) that are suggested to play major roles in the normal state of the iron chalcogenide FeSe [@YuSi_AFQ; @Lai2016; @Wenjun_nematic2016]. Indeed, at the semi-classical level, there are no corrections to the energies of these states from the ring exchanges.

From the above, we can conclude that the frustrated magnetism encoded in the Hamiltonian, Eq. , is sufficiently rich to understand the magnetism and nematic properties of FeTe on the same footing with those of FeSe and the iron pnictides. Given the strong indication for the importance of the spin physics to the iron-based superconductivity [@PCDai2015; @Si2016], the unification we have achieved here represents an important virtue of the mechanism we have advanced in the present work.

Finally, it is worth noting that alternative mechanisms for the BC in FeTe have assumed that the spin physics itself does not yield the ($\pi/2$,$\pi/2$) bicollinear antiferromagnetic state, but such an order arises under additional FeTe-specific driving forces beyond the spin physics. The additional driving forces that have been discussed 