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  "body": "\n## Summary\n\nThe Frölicher–Nijenhuis bracket is a graded Lie bracket on differential forms with values in the tangent bundle of a [[smooth manifold|F45VQ47GPV]].\nOn vector fields, which are the forms of degree $0$, it is the usual [[Lie bracket of vector fields|68FGQEBD6J]].\nIt is defined through the derivations of the algebra of [[differential forms|0E3G0GRPZ6]] that commute with the [[exterior derivative|0ZHSTAG8ZY]].\nIt expresses integrability conditions, such as those of almost complex structures and of connections, as the vanishing of a bracket.\n\n## Explanations\n\n### L1 human\n\nThe Lie bracket of two vector fields measures how far their flows fail to commute.\nVector-valued forms are more general objects, such as a field of linear maps on tangent spaces.\nThe Frölicher–Nijenhuis bracket extends the Lie bracket to all of them in a consistent way.\nFor example, for an almost complex structure $J$, the bracket of $J$ with itself measures whether $J$ comes from complex coordinates.\n",
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