Palatini identity

In general relativity and tensor calculus, the Palatini identity is:
where formula_2 denotes the variation of Christoffel symbols and formula_3 indicates covariant differentiation.
A proof can be found in the entry Einstein–Hilbert action.
The "same" identity holds for the Lie derivative formula_4. In fact, one has:
where formula_6 denotes any vector field on the spacetime manifold formula_7.