I am trying to compute the curvature for a gauge theory based on the pure (local) Lorentz group.
The final hurdle is working with monstrous structure constants.
My objective is to show that
\begin{eqnarray*} \Big[\big(\frac{1}{2}\omega_\mu{}^{ab}M_{ab}\big),\big(\frac{1}{2}\omega_\nu{}^{cd}M_{cd}\big)\Big] &=& \frac{1}{2} M_{ef}(\omega_\mu{}^{ec}\omega_{\nu c}{}^{f}-\omega_\nu{}^{ec}\omega_{\mu c}{}^{f}) \end{eqnarray*}
using the following commutation relations/structure constants:
The antisymmetric (non-Hermitean) $M_{ab}$ Lorentz generators obey the following commutation relation
$$[M_{ab},M_{cd}]=\eta_{bc}M_{ad}+\eta_{ad}M_{bc}-\eta_{ac}M_{bd}-\eta_{bd}M_{ac}$$
In analogy with the $[T^a,T^b]=if^{abc}T^c$ of Yang-Mills, the structure constants can be written out with some antisymmetrization deftness,
$$[M_a,M_b]=i f_{ab}{}^c M_c \rightarrow [M_{\mu\nu},M_{\rho\sigma}]=\frac{1}{2} f_{[\mu\nu][\rho\sigma]}{}^{[\kappa\tau]} M_{\kappa\tau}$$
where the structure constants are
$$ f_{[\mu\nu][\rho\sigma]}{}^{[\kappa\tau]} = 8 \eta_{[\rho[\nu}\delta^{[\kappa}_{\mu]}\delta^{\tau]}_{\sigma]}$$
I think the first here MAY be (although I'm not sure if this is allowed) \begin{eqnarray*} \Big[\big(\frac{1}{2}\omega_\mu{}^{ab}M_{ab}\big),\big(\frac{1}{2}\omega_\nu{}^{cd}M_{cd}\big)\Big] &=& \frac{1}{2} [M_{ab},M_{cd}] (\omega_\mu{}^{ab}\omega_\nu{}^{cd}-\omega_\nu{}^{cd}\omega_\mu{}^{ab}) \end{eqnarray*}
