 Solution

The relation isTaking its trace, symmetric traceless part, and antisymmetric part separately givesEquivalently,

If $\beta =-\gamma \ne 0$, $\alpha \ne 0$, and $A_{ij}=0$, its symmetric part forces $B_{kk}=0$ and its antisymmetric part forces $B_{[ij]}=0$. Thus the solutions are exactly the symmetric traceless tensors $B_{ij}$.