The _Lie bracket_ is a [bilinear map](Bilinear%20map.md) denoted as $[.,.],$ where for $X,$ $Y,$ and $Z$ the following properties are met: 1) $[X,Y]=-[Y,X].$ ([skew symmetry](Bilinear%20map.md#Symmetry%20properties%20of%20bilinear%20maps)) ^92bb0e 2) $[X,[Y,Z]] = [Y,[Z,X]] = [Z,[X,Y]] = 0$ ([The Jacobi identity](Jacobi%20identity.md)). ^f4cd08 # List of Lie brackets Here we list examples of Lie Brackets. * [operator commutators](Commutators.md) ^e15c43 * [cross product](Cross%20products.md) ^37c4ef * [Poisson bracket](Poisson%20bracket.md) # Ado's theorem [[Ado's Theorem]] #MathematicalFoundations/Geometry/DifferentialGeometry #MathematicalFoundations/Algebra/AbstractAlgebra/GroupTheory/Lie/LieGroups/Algebras