For a matrix Lie group, the left-invariant Maurer-Cartan form is . It identifies each tangent space with the Lie algebra and obeys .
When sums over ordered pairs, the Maurer-Cartan equation determines only . Symmetric additions vanish in the exterior product. The usual coefficients are the unique antisymmetric representatives; summing instead over removes the factor one half.
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The Maurer–Cartan form is a fundamental concept in the theory of Lie groups and differential geometry, particularly in the study of Lie group representations and the geometry of principal bundles. Given a Lie group \( G \), the Maurer–Cartan form is a differential 1-form on the Lie group that captures information about the group structure in terms of its tangent space.