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In the process of calculating a spin-averaged square amplitude in QFT, I came across the following expression: $$ \text{Tr}\left[\gamma^\mu\gamma^5\gamma^\alpha\gamma^\nu\gamma^5\gamma^\beta\right] $$ How can I evaluate this? The usual identities (eg the ones on Wikipedia) don't seem to be of much use - I can't see any way to do it with those, since there are two factors of $\gamma^5$.

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I just realised that I can use the fact that $\gamma^5$ anticommutes with the others (twice): $$=\text{Tr}\left[\gamma^\mu\gamma^\alpha\gamma^\nu\gamma^5\gamma^5\gamma^\beta\right]$$ then note that $\gamma^5\gamma^5$ is just the identity, giving: $$=\text{Tr}\left[\gamma^\mu\gamma^\alpha\gamma^\nu\gamma^\beta\right]$$ which can be expanded by the usual identity, resolving my issue.

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