The mechanical behavior of rock salt formations is not only of interest due to mining of salt but also because of its increasing relevance for geotechnical applications, e.g., for the storage of strategic oil and gas reserves, for hosting nuclear waste repositories, for compressed air energy storage (CAES) or the cavern storage of hydrogen. The safety assessment of geotechnical installations as well as long-term convergence predictions of salt caverns require models capturing the inelastic behavior of rock salt under thermomechanical loading. Many models have been derived for uniaxial or triaxial experimental setups and some are inadequately presented to fully describe three-dimensional scenarios. Here, a tensorial representation is chosen to avoid such inconsistencies and their possible consequences on commonly used material parameters are discussed. Two very commonly used material models, the LUBBY2 and Minkley formulations, have been primarily implemented into software relying on explicit time integration schemes. Here, their implicit implementation with analytical Jacobians into a scientific open-source finite element framework is described in detail. The implementations are verified by comparison to suitable analytical solutions for coupled thermomechanical loadings and their convergence behavior is analyzed.