Lie bracket
From Mathematics wiki
Given two sufficiently differentiable vector fields 
and 
,
on a manifold 
, the Lie bracket is defined as
![[\vec{u},\vec{v}\,] = \left(u^\beta \frac{\partial v^\alpha}{\partial x^\beta} - v^\beta \frac{\partial u^\alpha}{\partial x^\beta}\right) \frac{\partial}{\partial x^\alpha}\,](/w/images/math/3/8/d/38d518dccf6a1b1e1789498959f89a32.png)
.
It is related to the Lie derivative via
![[\vec{u},\vec{v}\,] = \mathcal{L}_\vec{u} \vec{v}\,](/w/images/math/2/f/3/2f3d7eb9659481ce12b397fc198b736b.png)
,
and has the following properties
![\left[\, , \,\right]\,](/w/images/math/9/9/a/99aa57ac15d9d06d1cbb41f36e7b1133.png)
is bilinear,
![[\vec{u},\vec{v}\,] = -[\vec{v},\vec{u}\,]\,](/w/images/math/4/6/7/4677ecfff344d4a4acb8df61842dde89.png)
,
![[\vec{u},[\vec{v},\vec{w}]]+[\vec{w},[\vec{u},\vec{v}]]+[\vec{v},[\vec{w},\vec{u}]]=0\,](/w/images/math/a/1/7/a17b1f66cf93c6598d31814f8e5fbb8b.png)
(Jacobi identity).
Venturing into differential geometry, if we endow 
with an inner product, i.e., a metric tensor, then the Lie bracket equips with the structure of a Lie algebra where the commutation coefficients play the role of the structure constants.
