# Evaluating a trace with two factors of $\gamma^5$

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 $$\gamma^5$$.

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.