Quaternionic modular forms (QMFs) are a type of non-holomorphic automorphic function that exist on certain forms of the exceptional groups, and on orthogonal groups SO(4,n) with n at least 3.  They have a robust notion of Fourier coefficients, defined in an analytic way.  Using the Fourier coefficients, I will give an algebraic characterization of the cuspidal quaternionic modular forms (on F_4, E_6, E_7, E_8) in terms of holomorphic modular forms on smaller rank groups.  That is, I will explain how cuspidal QMFs on these exceptional groups can be reconstructed from holomorphic modular forms on groups of type SO(2,n) and SL_2 in an algebraic way.  The main step in the proof is an "automatic convergence" result: An infinite sum that looks like it could be the Fourier expansion of a QMF actually does converge to an honest cusp form.  As a consequence, I deduce that the cuspidal QMFs on F_4, E_6, E_7, and E_8 have a basis where every element of this basis has all Fourier coefficients being algebraic numbers.