Summary: We strategically separate different core outcomes. The natural counterparts of a core allocation in a strategic environment are the \({\alpha}\)-core, the \({\beta}\)-core and the strong equilibrium, modified by assuming that utility is transferable in a strategic context as well. Given a core allocation \(\omega\) of a convex transferable utility (TU) game \(v\), we associate a strategic coalition formation game with \( \left( {v, \omega } \right) \) in which \(\omega\) survives, while most other core allocations are eliminated. If the TU game is strictly convex, the core allocations respected by the TU-\({\alpha}\)-core, the TU-\({\beta}\)-core and the TU-strong equilibrium shrink to \(\omega\) only in the canonical family of coalition formation games associated with \( \left( {v, \omega } \right) \). A mechanism, which strategically separates core outcomes from noncore outcomes for each convex TU game according to the TU-strong equilibrium notion is reported.