## Trace theorems and Dirac gamma matrices

Okay, first of all I need to define a representation of Dirac gamma matrices:
$\gamma ^0 = \left ( \begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{matrix} \right )$

$\gamma ^1 = \left ( \begin{matrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \end{matrix} \right )$

$\gamma ^2 = \left ( \begin{matrix} 0 & 0 & 0 & -i \\ 0 & 0 & i & 0 \\ 0 & i & 0 & 0 \\ -i & 0 & 0 & 0 \end{matrix} \right )$

$\gamma ^3 = \left ( \begin{matrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \\ -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \end{matrix} \right )$

And finally, we have the anticommutator: $\gamma ^{\mu} \gamma ^{\nu} + \gamma ^{\nu} \gamma ^{\mu} = 2 \eta ^{\mu \nu} I_4$
$\eta ^{ \mu \nu}$ is the usual Minkowski metric in SR. (I_4 is the 4 x 4 identity matrix. This is usually suppressed.)

To add a bit of complexity (though I think it makes things more illustrative), these form a 4-vector:
$\Gamma = \left ( \begin{matrix} \gamma ^0 \\ \gamma ^1 \\ \gamma ^2 \\ \gamma ^3 \end{matrix} \right )$

Okay, to the point. We can derive certain simplifications to traces. For example:
$Tr( \gamma ^{ \mu } \gamma ^{ \nu } ) = 2 \eta ^{\mu \nu}$

The trace of an odd number of gamma matrices is 0.

I know the formula for $Tr ( \gamma ^{ \mu } \gamma ^{ \nu } \gamma ^{ \lambda } \gamma ^{ \sigma } )$. My question is this: How do I find the trace of 8 gamma matrices? The problem seems to be rather forbidding.

Thanks!

-Dan