Recall that a vector field \(\boldsymbol{\vec v}\) acts on functions \(f\) by taking the directional derivative, that is \[\boldsymbol{\vec v}(f) = \boldsymbol{\vec v}\cdot\boldsymbol{\vec\nabla}f\]
Show that the set of all vector fields on \(\mathbb{R}^3\) forms a Lie algebra. That is, show first that the commutator of any two vector fields is a vector field then show that vector fields automatically satisfy the Jaobi identity.
In particular, this shows that the vector space of real linear combinations of any (finite) set of vector fields that is closed under commutation is a Lie algebra. (The set of all vector fields is an infinite-dimensional vector space over the reals.)