Showing that the Einstein Tensor has zero divergence

[kudoushinichi88]

Homework Statement

We have

$$R_{iklm;n}+R_{iknl;m}+R_{ikmn;l} \equiv 0$$

Show that by multiplying above with $g^{im}g^{kn}$

we'll get

$$\left( R^{ik}-\frac{1}{2} g^{ik} R \right)_{;k}$$


2. The attempt at a solution

$$g^{im}g^{kn} \left( R_{iklm;n}+R_{iknl;m}+R_{ikmn;l} \right) \equiv 0$$

$$g^{im}R_{i} ^{n}_{lm;n}+g^{kn}R^{m}_{knl;m}+\frac{\partial R}{\partial x^l} \equiv 0$$

$$R^{n}_{l;n}+R^{m}_{l;m}+\frac{\partial R}{\partial x^l} \equiv 0$$

then I'm stuck, not sure how to proceed. Honestly I'm not sure if my contractions are correct either. Please help?