I am just curious, why this question is put on hold Physics application of $SO(8)$ and Spin(8) triality, while other questions that have broader ends (a real open list) are left open.
As far as I know, SO(8) and Spin(8) triality have not been used in physics application other than one example I give in 1+1D BDI Time-Reversal invariant Topological Superconductor and 2+1D Z2-Ising-symmetric Topological Superconductor have ℤ8 classifications (see a related post here).
Other uses of SO(8) and Spin(8) triality may be in String Theory (but may not count as real-world physics), or possibly in disorder system or nuclear spectrum or Kondo effect (these I am just suspecting, we don't really know and it is certainly widely UN-known). They are not totally obvious examples. And there do not form a big list, if one wants to give precise use and Reference.
A side comment is that scientists and physicists learn by giving examples [don't you?]. I am not sure that what the far-insight to kill questions asking for giving meaningful examples, except that the purpose is trying to suppressing some users to use the site.