The Lie derivative of a general covariant $4$-tensor is given by $$\mathcal{L}_{K}R_{abcd} = K^{e}\nabla_{e}R_{abcd} + R_{ebcd}\nabla_{a}X^{e} + R_{aecd}\nabla_{b}X^{e} + R_{abed}\nabla_{c}X^{e} + R_{abce}\nabla_{d}X^{e}.$$

If the $(0,4)$ covariant tensor $R$ is the Riemann tensor and admits all the symmetries of the same, then by the theorem on the inheritance of symmetries, a Killing symmetry would imply an affine symmetry which in-turn would imply a curvature symmetry (assuming the no non-metricity and no torsion). Let $K$ be a vector field that preserves the Riemann tensor $R_{abcd}$ such that we have the condition of curvature collineation to hold, i.e., $$\mathcal{L}_{K}R_{abcd} = 0.$$ I now want to use this property of curvature collineation and derive the Killing identity: $\nabla_{a}\nabla_{b}K_{c} = R_{dabc}X^{d}$ but haven't been able to do so. Any help is appreciated.