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  "body": "\n## Summary\n\nThe Maurer–Cartan form of a [[Lie group|7JBRGCVJRB]] $G$ is a [[differential form|0E3G0GRPZ6]] of degree one on $G$ with values in its Lie algebra $\\mathfrak{g}$.\nIt translates each tangent vector of $G$ back to the identity by a left translation.\nIt is left-invariant and satisfies the Maurer–Cartan equation $d\\omega + \\tfrac12 [\\omega \\wedge \\omega] = 0$, which encodes the Lie bracket.\nIt is the model for connection forms and for Cartan's method of moving frames.\n\n## Explanations\n\n### L1 human\n\nAt each point $g$ of a Lie group there is a tangent space, and left translation by $g^{-1}$ moves it to the tangent space at the identity.\nThe Maurer–Cartan form does exactly this: it reads every velocity of the group as a vector in one fixed space, the Lie algebra.\nFor a group of matrices, a velocity $V$ at $g$ is read as $g^{-1} V$, so the form is written $g^{-1}\\,dg$.\nFor the circle group $e^{i\\theta}$, it is $e^{-i\\theta}\\, d(e^{i\\theta}) = i\\, d\\theta$.\n",
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