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  "body": "\n## Summary\n\nThe Lie bracket of two vector fields $X$ and $Y$ on a smooth manifold is a third vector field $[X, Y]$.\nIt measures how much the flows of $X$ and $Y$ fail to commute.\nIt is zero exactly when, for short times, moving along $X$ then $Y$ gives the same result as moving along $Y$ then $X$.\nWith this bracket the vector fields on a manifold form a Lie algebra.\nIt appears in the Frobenius theorem, in the definition of curvature and torsion, and in control theory.\n\n## Explanations\n\n### L1 human\n\nThink of a car: it can drive forward and it can turn the wheels, but it cannot slide sideways.\nBy alternating small forward and turning moves, you can still end up shifted sideways, as in parallel parking.\nThe sideways motion comes from Lie brackets of the motions you are allowed.\nOn the plane, the fields $\\partial_x$ and $x\\,\\partial_y$ have bracket $\\partial_y$, a direction neither field gives at the origin.\n",
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