Hecke modules associated to certain even ternary lattices of half-discriminant ${\mathbf{N}}^{\mathbf{2}}$

We consider Hecke modules of orthogonal modular forms attached to definite even ternary lattices of half-discriminant $N^2$, where $N>1$ is odd and square-free. We prove explicit Jacquet-Langlands style correspondences with spaces of classical modular forms (in the spirit of Birch and Hein-Tornaria-Voight). This is done by applying sequences of Watson-style transformations that reduce the discriminant while preserving the adelic orthogonal group.