Quantum many-body states of up to four excitons are studied in small parallelepipedal quantum dots with only two nondegenerate electronic levels in the conduction and in the valence bands. Only one type of hole is considered. We find the set of good quantum numbers (including spin variables for the electrons and holes) which block-diagonalizes the Coulomb and the optical dipole interactions. This provides a convenient labeling scheme for the excited states which allows us to diagonalize nearly entirely analytically the 256 x 256 Hamiltonian. The magnitude of the Coulomb level shifts and of the oscillator strength of the optical transitions are shown for a typical case.