Maurer-Cartan form

Definition

Let $G$ be a (connected) Lie group. The Maurer-Cartan form of $G$ is a 1-form on $G$ with values in the Lie algebra $\mathfrak{g}$, given as follows.

The tangent vector $v$ at a point $a\in G$, is sent to the unique vector $X\in \mathfrak{g}$ which is mapped by the differential of $g\mapsto ag$, to $v$. In other words, the map is:

$v\mapsto Df(v)$

where:

$f:=g\mapsto a^{-1}g$