Suppose I have some tensor, for concreteness I will consider a rank 4 tensor with components $R_{abcd}$ where the indices run over 0,1,2,3 (my question arises in a physics context). Under what conditions is the existence of another tensor, call it $G_{abcd}$, which satisfies $G_{ab}^{~~~mn}G_{cdmn}=R_{abcd}$ guaranteed?
Edit: Indices are raised and lowered using the metric tensor : $G_{ab}^{~~~mn}=g^{cm}g^{dn}G_{abcd}$. Also, for my problem $R_{abcd}$ is the Riemann tensor, and I suppose this would place extra conditions on the existence of a suitable $G_{abcd}$ since it is defined in terms of the derivatives of the metric (in the torsion free case). This probably complicates things quite some, so for now I will leave $R_{abcd}$ to be any rank 4 tensor.