The quaternionic Maass Spezialschar on split \(\mathrm{SO}(8)\)

The classical Maass Spezialschar is a Hecke-stable subspace of the level one holomorphic Siegel modular forms of genus two, i.e., on \(\mathrm{Sp}_4\), cut out by certain linear relations between the Fourier coefficients. It is a theorem of Andrianov, Maass, and Zagier, that the classical Maass Spezialschar is exactly equal to the space of Saito-Kurokawa lifts. We study an analogous space of quaternionic modular forms on split \(\mathrm{SO}_8\), and prove the analogue of the Andrianov-Maass-Zagier theorem. Our main tool for proving this theorem is the development of a theory of a Fourier-Jacobi expansion of quaternionic modular forms on orthogonal groups.