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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?

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  • $\begingroup$ For starters, $\theta$ should be $g^{-1}dg$. But why do you expect to get $0$? At the identity element of the group, you're getting $[dg,dg]$, and this is not $0$. Remember that $dg$ is $\mathfrak g$-valued, and $[dg(A),dg(B)] = [A,B]\ne 0$ for general elements $A,B\in\mathfrak g$. (Matrix-valued $1$-forms $\theta$ do not satisfy $\theta\wedge\theta = 0$.) $\endgroup$ – Ted Shifrin Oct 7 '18 at 22:43

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