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Given a smooth manifold $X$, the Lie bracket of vector fields $u$ and $v$ can be defined in several ways.
Since derivations of smooth functions are vector fields, we can identify $u$ and $v$ with the corresponding derivations $C^\infty(X)\to C^\infty(X)$.
Taking the commutator $uv-vu$ of these derivations produces another derivation, which is denoted by $[u,v]$, and which can be identified with a vector field on $X$.
Alternatively, we can set
$[u,v]=\mathcal{L}_u v=\mathcal{L}_v u,$where $\mathcal{L}$ denotes the Lie derivative of a vector field.
The real vector space of vector fields on $X$ equipped with the Lie bracket forms a Lie algebra.
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