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On page 172 of Schwatz’s QFT book, he derives the Klein–Gordon equation from Dirac equation as following:

$$(i \not\partial +m) (i \not\partial -m)\psi=(-\frac{1}{2} \partial_\mu \partial_\nu {\gamma^\mu \gamma^\nu}-\frac{1}{2} \partial_\mu \partial_\nu [\gamma^\mu \gamma^\nu]-m^2)\psi=-(\square +m^2)\psi =0$$

How does the second term that containing commutator of gamma matrices vanish?

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$\partial_\mu\partial_\nu$ is symmetric and $[\gamma_\mu,\gamma_\nu]$ is antisymmetric.

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