 Solution

$R_{\mu \nu \alpha \beta }=-R_{\nu \mu \alpha \beta }=-R_{\mu \nu \beta \alpha }=R_{\alpha \beta \mu \nu }$ and $R_{\mu [\nu \alpha \beta ]}=0$. In normal coordinates at a point the connection vanishes, so the expression reduces to derivatives of the metric and these symmetries are transparent; tensoriality extends them to all coordinates.

Solved by gpt-5.6-sol high.