i want to show that the following equation holds: $$ \frac{1}{8}\left[\gamma^{\mu},\omega_{\mu \nu} [\gamma^{\mu},\gamma^{\nu} ] \right] = \omega^{\mu}_{~~~\nu}\,~ \gamma^{\nu} $$
$\gamma^{\mu}$ are the Dirac matrices and $\omega $ is the part of the infinitesimal Lorentz transformation that is responsible for boosts and rotations. $$L^{\mu}_{~~~\nu} = \delta^{\mu}_{\nu}+ \omega^{\mu}_{~~~\nu} $$
I didn't get very far. I just replaced the commutator of the Dirac matrices with anticommutator $[\gamma^{\mu},\gamma^{\nu}]=2\gamma^{\mu}\gamma^{\nu}-2g^{\mu\nu}$ and ended up with: $$ \frac{1}{4}\left[ \gamma^{\mu}\omega_{\mu\nu}\gamma^{\mu}\gamma^{\nu}-\gamma^{\mu} \omega_{\mu\nu} \eta^{\mu\nu}-\omega_{\mu\nu}\gamma^{\mu}\gamma^{\nu}\gamma^{\mu} +\omega_{\mu\nu}\eta^{\mu\nu}\gamma^{\mu}\right] $$ for the left hand side.
Another idea was to use the leibniz rule for commutators $[A,BC]=[A,B]C+B[A,C]$: $$ \frac{1}{8}\left[\gamma^{\mu},\omega_{\mu \nu} [\gamma^{\mu},\gamma^{\nu} ] \right] =\frac{-i}{2}\left( \left\{ \gamma^{\mu},\omega_{\mu\nu}\right\} S^{\mu\nu}-\omega_{\mu\nu}\left\{\gamma^{\mu},S^{\mu\nu} \right\}\right)$$
Both ideas seem to fail since i don't know how to proceed. Can somebody please help me? Links and References to literature are welcome, too :-)
Thanks in Advance, mechanix