I figured it out, here it is
Using
\gamma^{\nu}\gamma^{\rho}=2g^{\nu\rho}-\gamma^{\rho}\gamma^{\nu}
we have
\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}=\gamma^{\mu}\left(2g^{\nu\rho}-\gamma^{\rho}\gamma^{\nu}\right)
=2\gamma^{\mu}g^{\nu\rho}-\gamma^{\mu}\gamma^{\rho}\gamma^{\nu}
next we...
here is what I have so far
\gamma^{5}\gamma^{\rho}=\frac{i}{24}\epsilon^{\sigma\mu\nu\rho}\gamma_{\sigma}\gamma_{\mu}\gamma_{\nu}\gamma_{\rho}\gamma^{\rho}=\frac{i}{6}\epsilon^{\sigma\mu\nu\rho}\gamma_{\sigma}\gamma_{\mu}\gamma_{\nu}...
Wiki identities and other identities are allowed, I am trying to start by expressing
\gamma^{5}=\frac{i}{24}\epsilon^{\mu\nu\rho\sigma}\gamma_{\mu}\gamma_{v}\gamma_{\rho}\gamma_{\sigma}
from which it follows that...
I need help proving the identity
\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}=\gamma^{\mu}g^{\nu\rho}+\gamma^{\rho}g^{\mu\nu}-\gamma^{\nu}g^{\mu\rho}+i\epsilon^{\sigma\mu\nu\rho}\gamma_{\sigma}\gamma^{5}