- Please let me know if I'm right about this:
I believe your question is equivalent to this question. Let S be the left-handed half-spinor representation of Spin(8) on 𝕆 and let V be the vector representation of Spin(8) on 𝕆. Is there an element g of Spin(8) such that V(g) preserves ℍ ⊂ 𝕆 while S(g) maps ℍ to its orthogonal complement?
Is that equivalent?
