In this talk, I will be discussing two classes of Bell diagonal indecomposable entanglement witnesses in C^4 ⊗ C^4. The first class is a generalization of the well-known Choi witness from C^3 ⊗ C^3 , while the second one contains the reduction map. I will show contrary to C^3 ⊗ C^3 case, the generalized Choi witnesses are no longer optimal. Thereafter, I will talk about an optimization procedure for finding spanning vectors that eventually give rise to optimal witnesses. Operators from the second class turn out to be optimal, however, without the spanning property. Our analysis sheds a new light into the intricate structure of optimal entanglement witnesses.
