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Let $(M^n,g)$ be a Riemannian manifold and let $W$ be its Weyl tensor. For a given ONB, does the identity $$W_{ijkl}W_{ijkm}=\frac{1}{n}|W|^2g_{lm}$$ hold? I think I've seen it somewhere but I'm not sure whether this is valid only in dimension four (in this case, this is certainly true).

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This does not hold for $n>4$. To see this, start with a Ricci-flat 4-manifold $N^4$ that is not flat, and let $M^{n}= N^4 \times \mathbb{R}^{n-4}$, endowed with the product metric. Then the metric on $M$ is Ricci-flat, so its Riemann curvature tensor is its Weyl tensor and $W_{ijkl}=0$ when any index is greater than $4$. However it is now clear that we can't have your equation when $l = m > 4$.

(This argument doesn't start to work by $n=4$ because the Weyl curvature is identically zero until you get to dimension $4$.)

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