is Spin(8) triality the missing ingredient for SUSY in Connes’ approach?

Thinking about a major puzzle in the NCG approach to the standard model, I remember that sci.physics.research, via Baez’ weeks, was very fond of triality (in the way of Evans?) to justify why some dimensions allow supersymmetry. And this pivots over SO(8), for which I asked a couple of abusive questions in mathoverflow:

Does ??(32)???8×?8 relate to some group theoretical fact?

Why SU(3) is not equal to SO(5)?

The second one included a nice ascii sequence of dynkin diagrams, using

o====o SO(5), isometries of the sphere S4o----o SU(3) are the isometries of CP2o o SU(2)xSU(2), isometries of S2xS2. Also SO(4), so isometries of S3

I compared

o o o / / o----o SO(8) o----o SU(3)xSO(4) o====o SO(5)xSO(4) \ \ o o o

and I wonder if I should add Pati-Salam

o o----o----o SU(4)xSU(2)xSU(2) o