With metric $(+---)$, $\gamma^5=i\gamma^0\gamma^1\gamma^2\gamma^3$ and $\epsilon^{0123}=+1$, odd ordinary <gamma matrix> traces vanish, $\operatorname{Tr}(\gamma^\mu\gamma^\nu)=4g^{\mu\nu}$, and
$$
\operatorname{Tr}(\gamma^\mu\gamma^\nu\gamma^\rho\gamma^\sigma)=4(g^{\mu\nu}g^{\rho\sigma}-g^{\mu\rho}g^{\nu\sigma}+g^{\mu\sigma}g^{\nu\rho}),\qquad \operatorname{Tr}(\gamma^5\gamma^\mu\gamma^\nu\gamma^\rho\gamma^\sigma)=-4i\epsilon^{\mu\nu\rho\sigma}.
$$
The first follows by repeated <Clifford algebra> anticommutation and cyclicity of the <trace>. The second is an antisymmetric tensor, whose normalization is fixed by the $0123$ component.


 Gamma matrix trace identities

ID: gamma-matrix-trace-identities