What exactly are the independent components?

Question

What are the 20 independent, non zero components of the 4D Riemann curvature tensor? (Not how many, I know there are twenty, but specifically which components are non-zero?)

Answer 1

It holds $R_{\alpha \alpha \mu \nu} = 0$ for every Dimension $\alpha$ where $R_{\lambda \kappa \mu \nu}$ takes derivatives in $\lambda$ and $\kappa$ direction due to antisymmetry. All other components of the Riemann Tensor are non-Zero.

Answer 2

When you consider the Riemann tensor as a map $R(p,q,r,s) = R_{abcd} p^a q^b r^c s^d$ for vectors $p,q,r,s$ in some basis (with summations implied), then some of the components $R_{abcd}$ are automatically zero: whenever $a=b$ or $c=d$ for instance.

This can seem a bit mystifying--why are those automatically zero? One way to look at it is that the Riemann tensor is a statement about how derivatives commute in planes--and if the derivatives don't commute, then in which planes do we pick up extra terms?

So the $ab$ or $cd$ pairs of indices are meant to represent planes, and those planes would be degenerate if $a=b$ or $c=d$. Hence, the Riemann tensor components there are automatically zero.