Spin-orbit effects on electron-electron interaction are studied theoretically. The corrections to the Coulomb interaction of quantum well electrons induced by the spin-orbit coupling are derived. The developed theory is applied to calculate the energy spectrum fine structure of an electron pair triplet states localized in small lateral disk-shaped quantum dots. We show that the spin degeneracy of a triplet state is completely lifted in anisotropic quantum dots. Isotropic quantum dots also demonstrate the peculiar fine structure of triplet states caused by the spin-orbit interaction.