The Riemann curvature tensor $R_{ijkl}$ satisfies several algebraic index symmetries:
- $R_{ijkl} = -R_{jikl} = -R_{ijlk}$
- $R_{ijkl} = R_{klij}$
- $R_{i[jkl]} = 0.$
I more or less understand how to interpret the first two identities: the first one says that the Riemann curvature tensor can be interpreted as a linear operator on the space of 2-forms $\Lambda^2(M)$, while the second one says that this linear operator is self-adjoint.
But I don't really know how to think about the third identity, the algebraic Bianchi identity. Does it have any physical intuition?
(This question asks the same thing about the second or "geometric" Bianchi identity.)