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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:
$\displaystyle[X,Y](f)=X(Y(f))Y(X(f)).$ 
Definition (Local coordinates) Suppose $X$ and $Y$ are vector fields on a smooth $n$dimensional manifold $M$, suppose $(x^{1},\ldots,x^{n})$ are local coordinates around some point $x\in M$, and suppose that in these local coordinates
$\displaystyle X(x)$  $\displaystyle=$  $\displaystyle X^{i}(x)\frac{\partial}{\partial x^{i}}\Big_{x},$  
$\displaystyle Y(x)$  $\displaystyle=$  $\displaystyle Y^{i}(x)\frac{\partial}{\partial x^{i}}\Big_{x}.$ 
Then the Lie bracket of the above vector fields is the locally defined vector field
$[X,Y](x)=X^{i}\frac{\partial Y^{j}}{\partial x^{i}}\frac{\partial}{\partial x^% {j}}\Big_{x}Y^{i}\frac{\partial X^{j}}{\partial x^{i}}\frac{\partial}{% \partial x^{j}}\Big_{x}.$ 
(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 $M$.

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

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

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