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 S4
o----o SU(3) are the isometries of CP2
o 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
```

```
```

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