= Spinor decomposition of gauge curvature
{title2=$F_{AA\prime BB\prime}=\epsilon_{AB}\varphi_{A\prime B\prime}+\epsilon_{A\prime B\prime}\varphi_{AB}$}
In complexified four-dimensional space-time, antisymmetry gives
$$
F_{AA'BB'}=\epsilon_{AB}\varphi_{A'B'}+\epsilon_{A'B'}\varphi_{AB},
$$
where 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 $\varphi_{A'B'}=0$. Equivalently $\pi^{A'}\pi^{B'}F_{AA'BB'}=0$ for every primed <two-component spinor> $\pi$, since a quadratic polynomial vanishing for every <two-component spinor> has all symmetric coefficients zero.
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