Maurer-Cartan 1-forms "commute"?
Let's Check for $\theta=gdg^{-1}$, $\chi=dg g^{-1}$
$$[\theta,\chi]=\theta \wedge \chi - (-1)^{1*1} \chi \wedge \theta =\theta \wedge \chi + \chi \wedge \theta = gdg^{-1}\wedge dg g^{-1} + dg g^{-1} \wedge g dg^{-1}$$ But $dg g^{-1}=-g dg^{-1}$ So $$[\theta,\chi]=2 dg\wedge dg^{-1} \neq 0 $$
Where i made a mistake?