The energy spectrum fine structure of triplet two-electron states in nanostructures is investigated theoretically. Spin-orbit interaction-induced terms in the effective Hamiltonian of the electron-electron interaction are derived for zinc blende lattice semiconductor systems: quantum wells and quantum dots. The effects of bulk and structural inversion asymmetry are taken into account. Simple analytical expressions describing the splittings of the two-electron states localized in a single quantum dot and in a lateral double quantum dot are derived. The spin degeneracy of triplet states is shown to be completely lifted by the spin-orbit interaction. An interplay of the conduction band spin splitting and the spin-orbit terms in the electron-electron interaction is discussed. The emission spectra of hot trions and of doubly charged excitons are calculated and are shown to reveal the fine structure of two-electron states.