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

Let $tilde{mathfrak{g}}$ 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 $W(Lambda) = U(tilde{mathfrak{g}}_1) v_Lambda$,
where $tilde{mathfrak{g}} = tilde{mathfrak{g}}_{−1} oplus tilde{mathfrak{g}}_0 oplus tilde{mathfrak{g}}_1$ is a $mathbb{Z}$-gradation of $tilde{mathfrak{g}}$ associated with a $mathbb{Z}$-gradation $mathfrak{g} = mathfrak{g}_{−1} oplusmathfrak{g}_0 oplus mathfrak{g}_1$.
Using vertex operator relations, we reduce the Poincaré-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 $W(Lambda)$.