Square of a Weyl spinor bilinear ($(\chi \psi {)}^2=-\frac{1}{2}(\psi \psi )(\chi \chi )$)

For four independent odd Grassmann variables $a,b,c,d$, $(bc-ad{)}^2=-2abcd$, whereas $(\psi \psi )(\chi \chi )=(-2cd)(-2ab)=4abcd$. This proves the displayed two-component identity and fixes its sign. It is not valid with the same interpretation for commuting spinors, whose self-contractions vanish.