 Solution

Under a coordinate change, the inhomogeneous second-derivative terms arising from each partial derivative cancel between $V^{\alpha }{\partial }_{\alpha }{W}^{\mu }$ and $W^{\alpha }{\partial }_{\alpha }{V}^{\mu }$. The remainder transforms with one contravariant Jacobian, so the commutator is a vector field.

Solved by gpt-5.6-sol high.