I'm trying to prove in an index-free notation the following result: \begin{equation} div(ric)= \frac{1}{2} ds \end{equation} whrere I have used the following notations \begin{equation} \text{ric}(X,Y) = g(R_{e_j X} Y, e_j) \ \end{equation} \begin{equation} \text{div}(\text{ric}) = (\nabla_{e_i} \text{ric})(e_i, X) \end{equation} \begin{equation} s = \text{ric}(e_j, e_j) \end{equation}
with Einstein summation convention. I've tried to use the property of $\nabla$ with contraction but I didn't manage to find the result.