Theorem 5

$$\gamma_\mu\gamma^\mu = 4 ,$$ (7.69)

$$\gamma_\mu \not{a} \gamma^\mu = a_\nu \gamma_\mu \gamma^\nu \gamma^\mu$$

$$= -a_\nu \gamma_\mu \gamma^\mu \gamma^\nu + 2a_\nu \gamma^\mu g^{\nu\mu}$$

$$= -4a_\nu \gamma^\nu + 2\not{a}$$

$$= -4\not{a} + 2\not{a}$$

$$= -2 \not{a} ,$$ (7.70)

$$\gamma_\mu \not{a} \not{b} \gamma^\mu = \gamma_\mu \not{a} b_\nu \gamma^\nu \gamma^\mu$$

$$= -\gamma_\mu \not{a} b_\nu \gamma^\mu \gamma^\nu + 2\gamma_\mu \not{a} b_\nu g^{\nu\mu}$$

$$= 2\not{a}\not{b} + 2\not{b}\not{a}$$

$$= 2a_\mu b_\nu (\gamma^\mu\gamma^\nu+\gamma^\nu\gamma^\mu)$$

$$= 4 a_\mu b_\nu g^{\mu\nu}$$

$$= 4 a\cdot b ,$$ (7.71)

This leads to

$$\gamma_\mu \not{a} \not{b} \not{c} \gamma^\mu = -2 \not{c} \not{b} \not{a} ,$$ (7.72)

$$\gamma_\mu \not{a} \not{b} \not{c} \not{d} \gamma^\mu = 2[ \not{d} \not{a} \not{b} \not{c} + \not{c} \not{b} \not{a} \not{d} ] .$$ (7.73)