The classical Spezialschar is the subspace of the space of holomorphic modular forms on  $\mathrm{Sp}_4(\mathbb{Z})$ whose Fourier coefficients satisfy a particular system of linear equations. An equivalent characterization of the Spezialschar can be obtained by combining work of Maass, Andrianov, and Zagier, whose work identifies the Spezialschar in terms of a theta-lift from $\widetilde{\mathrm{SL}_2}$. Inspired by work of Gan-Gross-Savin, Weissman and Pollack have developed a theory of modular forms on the split adjoint group of type D_4. In this setting we describe an analogue of the classical Spezialschar, in which Fourier coefficients are used to characterize those modular forms which arise as theta lifts from holomorphic forms on $\mathrm{Sp}_4(\mathbb{Z})$.