$$
\operatorname{Tr}(A+B) \equiv \operatorname{Tr}(A)+\operatorname{Tr}(B)
$$$$
\operatorname{Tr}(A B \ldots Y Z) \equiv \operatorname{Tr}(Z A B \ldots Y)
$$$$
\gamma^{\mu} \gamma^{\nu}+\gamma^{\nu} \gamma^{\mu} \equiv 2 g^{\mu \nu} I
$$
Take the trace of the above formula
$$
\operatorname{Tr}\left(\gamma^{\mu} \gamma^{\nu}\right)+\operatorname{Tr}\left(\gamma^{\nu} \gamma^{\mu}\right)=2 g^{\mu \nu} \operatorname{Tr}(I)
$$$$
\operatorname{Tr}(\gamma^\mu \gamma^\nu)= \operatorname{Tr}(\gamma^\nu\gamma^\mu)
$$$$
\operatorname{Tr}\left(\gamma^{\mu} \gamma^{\nu}\right)=4 g^{\mu \nu}
$$$$
\begin{aligned}
\operatorname{Tr}(\gamma \cdot a)(\gamma \cdot b) &=\operatorname{Tr}(\gamma \cdot b)(\gamma \cdot a) \\
&=\frac{1}{2} \operatorname{Tr} a_{\mu} b_{\nu}\left\{\gamma^{\mu}, \gamma^{\nu}\right\} \\
&=a \cdot b \operatorname{Tr} 1=4 a \cdot b
\end{aligned}
$$
- The trace of any odd number of $\gamma$ matrices are zero.
$$
\begin{aligned}
\operatorname{Tr}\left(\gamma^{\mu} \gamma^{\nu} \gamma^{\rho}\right) &=\operatorname{Tr}\left(\gamma^{5} \gamma^{5} \gamma^{\mu} \gamma^{\nu} \gamma^{\rho}\right) \\
&=\operatorname{Tr}\left(\gamma^{5} \gamma^{\mu} \gamma^{\nu} \gamma^{\rho} \gamma^{5}\right) \\
&=-\operatorname{Tr}\left(\gamma^{5} \gamma^{5} \gamma^{\mu} \gamma^{\nu} \gamma^{\rho}\right)\\
&=-\operatorname{Tr}(\gamma^\mu\gamma^\nu \gamma^\rho)
\end{aligned}
$$$$
\operatorname{Tr}\left(\gamma^{\mu} \gamma^{\nu} \gamma^{\rho}\right)=0
$$$$
\begin{aligned}
\operatorname{Tr} a\llap{/}_1 \ldots a\llap{/}_{n} &=\operatorname{Tr} a\llap{/}_{1} \ldots a\llap{/}_{n} \gamma_{5} \gamma_{5} \\
&=\operatorname{Tr} \gamma_{5} a\llap/_{1} \ldots a\llap{/}_{n} \gamma_{5}
\end{aligned}
$$$$
\operatorname{Tr} a\llap/_{1} \ldots a\llap/_{n}=(-1)^{n} \operatorname{Tr} a\llap/_{1} \ldots a\llap/_{n} \gamma_{5} \gamma_{5}
$$$$
\operatorname{Tr} a\llap/_{1} \ldots a\llap/_{n}=0, \quad n \text { odd } .
$$
- Trace of even number of matrices
$$
\begin{aligned}
\gamma^{\mu} \gamma^{\nu} \gamma^{\rho} \gamma^{\sigma} &=2 g^{\mu \nu} \gamma^{\rho} \gamma^{\sigma}-\gamma^{\nu} \gamma^{\mu} \gamma^{\rho} \gamma^{\sigma} \\
&=2 g^{\mu \nu} \gamma^{\rho} \gamma^{\sigma}-2 g^{\mu \rho} \gamma^{\nu} \gamma^{\sigma}+\gamma^{\nu} \gamma^{\rho} \gamma^{\mu} \gamma^{\sigma} \\
&=2 g^{\mu \nu} \gamma^{\rho} \gamma^{\sigma}-2 g^{\mu \rho} \gamma^{\nu} \gamma^{\sigma}+2 g^{\mu \sigma} \gamma^{\nu} \gamma^{\rho}-\gamma^{\nu} \gamma^{\rho} \gamma^{\sigma} \gamma^{\mu}
\end{aligned}
$$$$
\Rightarrow \quad \gamma^{\mu} \gamma^{\nu} \gamma^{\rho} \gamma^{\sigma}+\gamma^{\nu} \gamma^{\rho} \gamma^{\sigma} \gamma^{\mu}=2 g^{\mu \nu} \gamma^{\rho} \gamma^{\sigma}-2 g^{\mu \rho} \gamma^{\nu} \gamma^{\sigma}+2 g^{\mu \sigma} \gamma^{\nu} \gamma^{\rho} .
$$$$
\operatorname{Tr}\left(\gamma^{\mu} \gamma^{\nu} \gamma^{\rho} \gamma^{\sigma}\right)=4 g^{\mu \nu} g^{\rho \sigma}-4 g^{\mu \rho} g^{\nu \sigma}+4 g^{\mu \sigma} g^{\nu \rho}
$$$$
\begin{aligned}
\operatorname{Tr}(\gamma \cdot a)(\gamma \cdot b)(\gamma \cdot c)(\gamma \cdot d)=&-\operatorname{Tr}(\gamma \cdot b)(\gamma \cdot a)(\gamma \cdot c)(\gamma \cdot d) \\
&+2 a \cdot b \operatorname{Tr}(\gamma \cdot c)(\gamma \cdot d)
\end{aligned}
$$
(a) $\operatorname{Tr}(I)=4$
(b) the trace of any odd number of $\gamma$-matrices is zero;
(c) $\operatorname{Tr}\left(\gamma^{\mu} \gamma^{\nu}\right)=4 g^{\mu \nu}$;
(d) $\operatorname{Tr}\left(\gamma^{\mu} \gamma^{\nu} \gamma^{\rho} \gamma^{\sigma}\right)=4 g^{\mu \nu} g^{\rho \sigma}-4 g^{\mu \rho} g^{\nu \sigma}+4 g^{\mu \sigma} g^{\nu \rho}$;
(e) the trace of $\gamma^{5}$ multiplied by an odd number of $\gamma$-matrices is zero;
(f) $\operatorname{Tr}\left(\gamma^{5}\right)=0$
(g) $\operatorname{Tr}\left(\gamma^{5} \gamma^{\mu} \gamma^{\nu}\right)=0$; and
(h) $\operatorname{Tr}\left(\gamma^{5} \gamma^{\mu} \gamma^{\nu} \gamma^{\rho} \gamma^{\sigma}\right)=4 i \varepsilon^{\mu \nu \rho \sigma}$, where $\varepsilon^{\mu \nu \rho \sigma}$ is antisymmetric under the interchange of any two indices.