 Global form obstruction for the Spin7 spinor

The Spin group $\mathrm{Spin}(7)$ 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 $\mathrm{Spin}(6)\cong SU(4)$ the same obstruction is visible because $-I_4$ acts trivially on ${\Lambda }^{2}4$ but negatively on $4\oplus \overline{4}$.