= Two-component spinor
{title2=$\pi_{A\prime}\in\mathbb C^2$}
A two-component <two-component spinor> is an element of a two-dimensional complex spin space with invariant antisymmetric form $\epsilon_{AB}$. In complexified four-dimensional geometry there are independent unprimed and primed spin spaces, and vectors are represented by $x^{AA'}$. Raising an index uses the antisymmetric form, so $\pi^{A'}\pi_{A'}=0$. This identity makes the incidence-plane directions $\pi^{A'}\partial_{AA'}$ totally null.
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