# Why is $\omega([X,Y])=[\omega(X),\omega(Y)]$ for left-invariant vector fields $X,Y$ and the Maurer-Cartan form $\omega$?

In other words, why is the Maurer-Cartan form $\omega$ a Lie algebra homomorphism between $\mathfrak g=T_eG$ and the Lie algebra of left-invariant vector fields on $G$? Why does it preserve the Lie bracket?

• See here under "Properties". By definition of the Lie bracket. – Dietrich Burde Sep 29 '16 at 21:09
• @DietrichBurde The Wiki article just says If X and Y are both left-invariant, then $\omega([X,Y])=[\omega(X),\omega(Y)]$. I don't think that really explains it. If you take this equality as the definition of the Lie bracket on $\mathfrak g$ (as the Wiki article seems to suggest), then how do you show that this definition is equal to other definitions, for example the standard definition in matrix Lie algebras like $\mathfrak{gl}(n)$, where the definition is $[X,Y]=XY-YX$? – Andy Miles Oct 28 '16 at 20:37
• Yes, you are right, Wiki suggests it. But then we have already proved why the Lie bracket for $\mathfrak{gl}(n)$ is given by $[X,Y]=XY-YX$, see here. – Dietrich Burde Oct 28 '16 at 20:41
• @DietrichBurde Ah I see. The Maurer-Cartan form is just the inverse of the isomorphism that maps Lie algebra elements $X$ to left-invariant vector fields $X$. Because that's a Lie algebra isomorphism, the Maurer-Cartan form must preserve the Lie bracket, right? – Andy Miles Oct 28 '16 at 20:45