High Quality Content by WIKIPEDIA articles! In the mathematical field of differential topology, the Lie bracket of vector fields, Jacobi–Lie bracket, or Commutator of vector fields is a bilinear differential operator which assigns, to any two vector fields X and Y on a smooth manifold M, a third vector field denoted [X, Y].For a matrix Lie group, smooth vector fields can be locally represented in the corresponding Lie algebra. Since the Lie algebra associated with a Lie group is isomorphic to the group's tangent space at the identity, elements of the Lie algebra of a matrix Lie group are also matrices.The Jacobi–Lie bracket is essential to proving small-time local controllability (STLC) for driftless affine control systems.

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