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In complexified four-dimensional space-time, antisymmetry giveswhere both gauge curvature two-component spinors are symmetric. The two summands are the two duality components. In the convention where the primed symmetric part is self-dual, the Anti-self-dual Yang-Mills equations are ${\phi }_{A^{\prime }{B}^{\prime }}=0$. Equivalently ${\pi }^{A^{\prime }}{\pi }^{B^{\prime }}{F}_{A{A}^{\prime }B{B}^{\prime }}=0$ for every primed two-component spinor $\pi$, since a quadratic polynomial vanishing for every two-component spinor has all symmetric coefficients zero.

A two-component two-component spinor is an element of a two-dimensional complex spin space with invariant antisymmetric form ${ϵ}_{AB}$. In complexified four-dimensional geometry there are independent unprimed and primed spin spaces, and vectors are represented by $x^{A{A}^{\prime }}$. Raising an index uses the antisymmetric form, so ${\pi }^{A^{\prime }}{\pi }_{A^{\prime }}=0$. This identity makes the incidence-plane directions ${\pi }^{A^{\prime }}{\partial }_{A{A}^{\prime }}$ totally null.