the Maurer-Cartan form
/ MOW-rer kar-TAHN /
On a Lie group there is one differential form so natural it needs no choices: at every point it reads off the velocity of a curve and, by sliding back to the identity, reports the answer as a single element of the Lie algebra. It is the universal 'gauge' that records motion on the group intrinsically. This is the Maurer-Cartan form, and its structure equation is the differential-form incarnation of the bracket.
Concretely, the (left) Maurer-Cartan form theta of a Lie group G is a g-valued 1-form on G: at a point g it sends a tangent vector v in T_g G to theta_g(v) = (L_(g^(-1)))_* v in g = T_e G — that is, left-translate the vector back to the identity. By construction theta is left-invariant and is an isomorphism T_g G -> g at every point (it 'trivializes' the tangent bundle). For a matrix group the formula is the clean expression theta = g^(-1) dg, where dg is the matrix of differentials of the entries. The central fact is the Maurer-Cartan structure equation: d theta + (1/2)[theta ^ theta] = 0, equivalently d theta + theta ^ theta = 0 in matrix notation, where [theta ^ theta] combines the wedge product of forms with the Lie bracket. This single equation packages the entire local structure of the group; the bracket reappears as the obstruction to theta being closed.
The Maurer-Cartan form is the seed of Cartan's method of moving frames and of the theory of connections on principal bundles (a connection is, locally, a g-valued 1-form whose restriction to fibers is the Maurer-Cartan form). Its structure equation is the flat, 'background' model whose curvature is exactly zero; curvature in the bundle theory measures the failure of a connection to satisfy the pure Maurer-Cartan equation. An honesty note: there are two Maurer-Cartan forms, the left one (g^(-1) dg) and the right one (dg g^(-1)); conventions differ, and the sign in the structure equation depends on the bracket-and-wedge convention, so always pin down which form and which convention is in force.
For a matrix group, theta = g^(-1) dg. Differentiate using d(g^(-1)) = -g^(-1) (dg) g^(-1): then d theta = d(g^(-1)) ^ dg = -g^(-1) dg ^ g^(-1) dg = -theta ^ theta, which is exactly the structure equation d theta + theta ^ theta = 0. The whole local Lie structure falls out of one matrix manipulation.
From theta = g^(-1)dg, a one-line matrix computation yields the Maurer-Cartan structure equation.
There are two conventions: the left form g^(-1)dg and the right form dg g^(-1), with sign differences in the structure equation. The MC form is the FLAT model — its 'curvature' is zero by construction; do not read it as a connection with curvature.