Generating sets of standard modules for $D_4^{(1)}$

Let $\widetilde{\mathfrak g}$ be an affine Lie algebra of type $D_4^{(1)}$ and $L(\Lambda)$ its standard module of level $k$ with highest weight vector $v_{\Lambda}$. We define Feigin--Stoyanovsky's type subspace as $W(\Lambda)=U(\widetilde{\mathfrak g}_{1})\,v_{\Lambda}$, where $\widetilde{\mathfrak g}=\widetilde{\mathfrak g}_{-1}\oplus\widetilde{\mathfrak g}_{0}\oplus\widetilde{\mathfrak g}_{1}$ is a $\mathbb{Z}$-gradation of $\widetilde{\mathfrak g}$ associated with a $\mathbb{Z}$-gradation $\mathfrak g=\mathfrak g_{-1}\oplus\mathfrak g_{0}\oplus\mathfrak g_{1}$. Using vertex operator relations, we reduce the Poincar\'e--Birkhoff--Witt spanning set of $W(\Lambda)$, and describe it in terms of difference and initial conditions. The spanning set of the whole standard module $L(\Lambda)$ can be obtained as a limit of the spanning set for $W(\Lambda)$.