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

Question

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?