Maurer–Cartan equation lemma

A computational identity: the exterior derivative of the Maurer–Cartan form is the negative bracket.

Maurer–Cartan equation lemma

Let $G$ be a Lie group with Lie algebra $\mathfrak g$ , and let $\theta$ be the left Maurer–Cartan form .

Lemma

For any smooth vector fields $X,Y$ on $G$ ,

(d\theta)(X,Y) = -[\theta(X),\theta(Y)].

Equivalently,

d\theta + \frac12[\theta,\theta]=0,

which is the Maurer–Cartan equation .

Context

This lemma is the workhorse behind computations with invariant forms and is the differential-geometric source of the Lie bracket, complementary to the flow-based viewpoint via one-parameter subgroups as integral curves .