7.2 Lie Algebra

The Lie algebra $\mathrm{\mathfrak{sl}}(3)$ has eight generators, all with zero trace:

$\displaystyle G_{1}={\scriptstyle \left(\begin{array}{ccc} 0 & 0 & 1\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{array}\right)},$	$\displaystyle G_{2}={\scriptstyle \left(\begin{array}{ccc} 0 & 0 & 0\\ 0 & 0 & 1\\ 0 & 0 & 0 \end{array}\right)},$	$\displaystyle G_{3}={\scriptstyle \left(\begin{array}{ccc} 0 & -1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 0 \end{array}\right)},$	(85)
$\displaystyle G_{4}={\scriptstyle \left(\begin{array}{ccc} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & -2 \end{array}\right)},$	$\displaystyle G_{5}={\scriptstyle \left(\begin{array}{ccc} 1 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & 0 \end{array}\right)}$	$\displaystyle G_{6}={\scriptstyle \left(\begin{array}{ccc} 0 & 1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 0 \end{array}\right)},$	(86)
$\displaystyle G_{7}={\scriptstyle \left(\begin{array}{ccc} 0 & 0 & 0\\ 0 & 0 & 0\\ 1 & 0 & 0 \end{array}\right)},$		$\displaystyle G_{8}={\scriptstyle \left(\begin{array}{ccc} 0 & 0 & 0\\ 0 & 0 & 0\\ 0 & 1 & 0 \end{array}\right)}$	(87)