Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 18; 8.
The second absolute gradient (or covariant derivative) of a four-vector is not commutative, as we can show by a direct derivation. Starting with the formula for the absolute gradient of a four-vector:
and the formula for the absolute gradient of a mixed tensor:
we can write out the second absolute gradient of a four-vector:
If we now swap and , we get, using the commutativity of ordinary derivatives and the symmetry of :
Subtracting these two equations gives
Using the definition of the Riemann tensor:
Thus the covariant derivative commutes only if the Riemann tensor is zero, which occurs only in flat spacetime.