I'm trying to grab some Riemannian geometry through "The geometry of Kerr black holes". After a few computations, I managed to derive the Bianchi curvature identities from the definition of the Riemann tensor and its cyclic symmetry.
This leads me to ask the following questions:
1) as the properties of the covariant derivative seem to play no role in this derivation, does it mean that the Bianchi identities hold for every tensor of type $(1,3)$ having the symmetries of the Riemann tensor?
2) is the Riemann tensor determined up to a multiplicative constant by its group of symmetries?
1) Yes, but good luck finding another interesting example of such a tensor.
2) Not in $4$ or more dimensions. (For $2$ or $3$, see here.)