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I know some Lie and representation theory, mostly from physics books. As I understand there are vector and spinor representations for so(n). If n is even, there are two spin reps and one vec rep, if n is odd there is one spin rep and one vec rep.

For example, for n=4 there are the two 2-dim Weyl spinors and the one 4-dim vec representation.

so(8) is special, since the dim of the two spin reps and the one vec rep is ithe same, 8.

Now, how do these two spinors look for so(8)?

They say triality gives the symmetry of Spin(8), not of so(8), it is an automorphism of Spin(8). What does that mean?

Thank you