[@P2], Primc found combinatorial basis of $W(\Lambda)$ for all classical Lie algebras and any possible gradation but only for basic modules $L(\Lambda_0)$. In this paper, he applied crystal base character formula ([@KKMMNN]) for level one modules but level $k > 1$ modules were not discussed.

Using the Caparelli-Lepowsky’s-Milas’ approach via intertwining operators and the description of the basis from [@FJLMM], Primc gave in [@P3] a simpler proof of linear independence for the basis of $W(\Lambda)$ constructed in [@P1]. In the similar manner, G. Trupčević obtained in [@T1] and [@T2] combinatorial bases of Feigin-Stoyanovsky’s type subspaces $W(\Lambda)$ for all standard modules of any level or gradation for $\mathfrak{sl}(\ell + 1, {\mathbb{C}})$ whilst M. Jerković managed to obtain the systems of recurrence relations for formal characters of Feigin-Stoyanovsky’s type subspaces of level $k$ standard modules for $\mathfrak{sl}(\ell + 1, {\mathbb{C}})^{\widetilde{}}$ ([@J]).

In this paper we extend the same approach to construct a combinatorial basis for Feigin-Stoyanovsky’s type subspace of four standard level 1 modules for algebra of type $D_{\ell}^{(1)}$. Then we construct a combinatorial basis for Feigin-Stoyanovsky’s type subspace of the level 2 standard modules for $D_{4}^{(1)}$ type algebra.

Let ${\mathfrak{g}}$ be a simple Lie algebra, ${\mathfrak{h}}$ a Cartan subalgebra of ${\mathfrak{g}}$ 