Combinatorial bases of Feigin-Stoyanovsky's type subspaces of level 2 standard modules for $D_4^{(1)}$

Let $\gtl$ be an affine Lie algebra of type $D_{\ell}^{(1)}$ and $L(\Lambda)$ its standard module with a highest weight vector $v_{\Lambda}$. For a given $\Z$-gradation $\gtl = \gtl_{-1} + \gtl_0 + \gtl_1$, we define Feigin-Stoyanovsky's type subspace as $$W(\Lambda) = U(\gtl_1) \cdot v_{\Lambda}.$$ By using vertex operator relations for standard modules we reduce the Ponicar\'{e}-Brikhoff-Witt spanning set of $W(\Lambda)$ to a basis and prove its linear independence by using Dong-Lepowsky intertwining operators.