Maurer–Cartan form

For a Lie group $G$ with Lie algebra $\mathfrak{g}$, the left Maurer-Cartan form is the $\mathfrak{g}$-valued differential 1-formIt translates a tangent vector at any point back to the identity, giving a linear isomorphism $T_{g}G\to \mathfrak{g}$. In particular, $\theta (X^L)=X$ for the left-invariant vector field determined by $X$. This makes $\theta$ left-invariant and supplies a global coframe after choosing a basis of $\mathfrak{g}$.

The exterior derivative on left-invariant vector fields gives $d\theta (X^L,Y^L)=-[X,Y]$, since the two derivative terms of constant coefficients vanish. Thus $d\theta +\frac{1}{2}[\theta ,\theta ]=0$, the Maurer-Cartan equation. For a Matrix Lie group, $\theta =g^{-1}dg$ and this becomes $d\theta +\theta \wedge \theta =0$, where the exterior product includes matrix multiplication. The additive real numbers have $\theta =dx$, while the positive multiplicative real numbers have $\theta =da/a$. For the real affine group, the Maurer-Cartan coframe of the real affine group makes the nonzero Lie bracket visible as $d(db/a)=-(da/a)\wedge (db/a)$.