Lie bracket
The Lie bracket is an anticommutative, bilinear^{}, first order differential operator on vector fields. It may be defined either in terms of local coordinates or in a global, coordinatefree fashion. Though both defintions are prevalent, it is perhaps easier to formulate the Lie Bracket without the use of coordinates^{} at all, as a commutator:
Definition (Global, coordinatefree) Suppose $X$ and $Y$ are vector fields on a smooth manifold $M$. Regarding these vector fields as operators on functions, the Lie bracket is their commutator:
$[X,Y](f)=X(Y(f))Y(X(f)).$ 
Definition (Local coordinates) Suppose $X$ and $Y$ are vector fields on a smooth $n$dimensional manifold $M$, suppose $({x}^{1},\mathrm{\dots},{x}^{n})$ are local coordinates around some point $x\in M$, and suppose that in these local coordinates
$X(x)$  $=$  ${{X}^{i}(x){\displaystyle \frac{\partial}{\partial {x}^{i}}}}_{x},$  
$Y(x)$  $=$  ${{Y}^{i}(x){\displaystyle \frac{\partial}{\partial {x}^{i}}}}_{x}.$ 
Then the Lie bracket of the above vector fields is the locally defined vector field
$$[X,Y](x)={{X}^{i}\frac{\partial {Y}^{j}}{\partial {x}^{i}}\frac{\partial}{\partial {x}^{j}}}_{x}{{Y}^{i}\frac{\partial {X}^{j}}{\partial {x}^{i}}\frac{\partial}{\partial {x}^{j}}}_{x}.$$ 
(The Einstein summation convention employed in the above equations — repeated indices are to be summed from the range 1 to $n$.)
Properties
Suppose $X,Y,Z$ are smooth vector fields on a smooth manifold $M$.

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$[X,Y]={\mathcal{L}}_{X}Y$ where ${\mathcal{L}}_{X}Y$ is the Lie derivative^{}.

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$[\cdot ,\cdot ]$ is antisymmetric and bilinear.

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Vector fields on $M$ with the Lie bracket is a Lie algebra. That is to say, the Lie bracket satisfies the Jacobi identity:
$$[X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]]=0.$$ 
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The Lie bracket is covariant with respect to changes of coordinates.
Title  Lie bracket 

Canonical name  LieBracket 
Date of creation  20130322 14:10:02 
Last modified on  20130322 14:10:02 
Owner  rspuzio (6075) 
Last modified by  rspuzio (6075) 
Numerical id  10 
Author  rspuzio (6075) 
Entry type  Definition 
Classification  msc 5300 
Related topic  HamiltonianAlgebroids 