I have seen the Ricci identity written variously as
- $R_{ijk}{}^l x^k = (\nabla_i\nabla_j- \nabla_j\nabla_i) x^l$
- $R_{ij}{}^l{}_k x^k = (\nabla_i\nabla_j- \nabla_j\nabla_i) x^l$
- $R^l{}_{kij} x^k = (\nabla_i\nabla_j- \nabla_j\nabla_i) x^l$
To my understanding 2. and 3. are equivalent but 1. has the opposite sign due to the symmetries of the Riemann tensor. Oddly I do not recall ever seeing the 4 possible variant.
Which is the correct form?