= Global form obstruction for the Spin7 spinor
{title2=$\mathbf8\text{ does not descend to }SO(7)$}
The <Spin group> $\operatorname{Spin}(7)$ double covers <SO(7)>. Its central kernel element $-1$ acts trivially in the vector <representation> and as $-I$ in the spinor <representation>. Thus a spinor <Lie algebra representation> does not define an ordinary <SO(7)> <group representation>. Under $\operatorname{Spin}(6)\cong SU(4)$ the same obstruction is visible because $-I_4$ acts trivially on $\Lambda^2\mathbf4$ but negatively on $\mathbf4\oplus\overline{\mathbf4}$.
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