A method for determining the elastic constants of low-symmetry crystal structures with internal coordinates through the computation of the Hellmann-Feynman forces is described. The approach applies to cases where previous methods were impractical, namely cases where the macroscopic strains are coupled to the internal degrees of freedom. The approach is illustrated for triclinic CaP3, which has 12 internal degrees of freedom. In non-cubic structures invariances must be imposed on the force constants, regardless of whether those interatomic force constants were obtained from a supercell-or from a linear-response approach. Moreover, it is shown that important aspects of crystal bonding are easily analyzed in terms of the force constants.