Maurer–Cartan form

Summary

The Maurer–Cartan form of a Lie group $G$ is a differential form of degree one on $G$ with values in its Lie algebra $\mathfrak{g}$. It translates each tangent vector of $G$ back to the identity by a left translation. It is left-invariant and satisfies the Maurer–Cartan equation $d\omega + \tfrac12 [\omega \wedge \omega] = 0$, which encodes the Lie bracket. It is the model for connection forms and for Cartan's method of moving frames.