Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 3 / 37D / a / ii / Solution

Every pair of <coordinate basis> fields commutes:
$$
[\partial_i,\partial_j]=0.
$$
Therefore a frame containing two fields with nonzero <Lie bracket of vector fields> cannot be induced by a coordinate system.

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 Incoming links: Commuting coordinate basis