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.