Theorem 2

$\displaystyle \textrm{Tr}[\not{a}\not{b}] = \textrm{Tr}[\not{b}\not{a}]$	$\textstyle =$	$\displaystyle \frac{1}{2} \textrm{Tr}[\not{a}\not{b} + \not{b}\not{a}] ,$
	$\textstyle =$	$\displaystyle \frac{1}{2} \textrm{Tr} [a_\mu b_\nu(\gamma^\mu\gamma^\nu+\gamma^\nu\gamma^\mu)] ,$
	$\textstyle =$	$\displaystyle \frac{1}{2} \textrm{Tr}[a_\mu b_\nu(2g^{\mu\nu})] ,$
	$\textstyle =$	$\displaystyle \textrm{Tr}[a\cdot b] ,$
	$\textstyle =$	$\displaystyle a\cdot b \textrm{Tr}[I] ,$
	$\textstyle =$	$\displaystyle 4 a\cdot b .$	(7.59)

$\displaystyle \textrm{Tr}[a_\mu\gamma^\mu b_\nu\gamma^\nu]$	$\textstyle =$	$\displaystyle 4a_\mu b^\mu ,$
$\displaystyle a_\mu b_\nu \textrm{Tr}[\gamma^\mu\gamma^\nu]$	$\textstyle =$	$\displaystyle 4a_\mu b_\nu g^{\mu\nu} ,$
$\displaystyle \Rightarrow \textrm{Tr}[\gamma^\mu\gamma^\nu]$	$\textstyle =$	$\displaystyle 4g^{\mu\nu} .$	(7.60)