I am currently reading the original paper by Chern and Simons where they introduce their form. I am working out the examples, but I do not seem to be able to derive their formula. We should be able to derive: $$TP_1 (\theta) = \frac{1}{4 \pi ^2} (\theta_{12} \wedge \theta_{13}\wedge \theta_{23}+{\theta}_{12}\wedge\Omega_{12}+\theta_{13}\wedge\Omega_{13}+\theta_{23}\wedge\Omega_{23}) $$ The 3 Chern-Simons form for a 3-manifold. We should only need the following equation applied to the case $l=2$ (since 3=2(2)-1): $$TP(\theta) = \sum_{i=0}^{l-1} A_i P(\theta \wedge [\theta,\theta]^i\wedge\Omega^{l-i-1}) $$ $$ A_i = (-1)^i l! \frac{(l-1)!}{2^i (l+i)!} (l-1-i)!$$
My work: I managed to reduce the formula to: $$ TP_1(\theta)=P(\theta \wedge \Omega) - \frac{1}{6} P(\theta \wedge [\theta,\theta])$$ How do I express this last part in the local frame to get the desired formula? Also, I will be interested in calculating the higher degree forms. How do I work out the power of bracket and so?