Examples
For a Lie group, the Lie algebra is the tangent space at the identity, which can be identified with the left invariant vector fields. The Lie bracket of the Lie algebra is then the Lie bracket of the left invariant vector fields, which is also left invariant.
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. Hence the Jacobi–Lie bracket corresponds to the usual commutator for a matrix group:
where juxtaposition indicates matrix multiplication.
Read more about this topic: Lie Bracket Of Vector Fields
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