${\tilde{\mathfrak{g}}}_1$ and send certain vectors from one $W(\Lambda^1)$ subspace to another $W(\Lambda^2)$ subspace. This operators will be the specific coefficients of the intertwining operators (see [@DL]). We also use the operator $e(\lambda)$ which has the property to lift the degree of the monomial (cf. [@DLM]), i.e. $$\begin{aligned}
 \label{operatoric}
x_{\alpha}(n)e(\lambda)=e(\lambda)x_{\alpha}(n + \langle \lambda, \alpha\rangle), \qquad n \in {\mathbb{Z}}.\end{aligned}$$ Using the intertwining operators and the operator we prove the linear independence by induction on the degree of the monomial on all subspaces of level 1 modules simultaneously.

In Sections \[pocetak2\] - \[kraj2\] we study Feigin-Stoyanovsky’s type subspaces of $L(\Lambda)$ where $L(\Lambda)$ is a fundamental level two module for algebra of type $D_4^{(1)}$. We also describe the monomials that satisfy the difference and initial conditions on such $W(\Lambda)$. They span $W(\Lambda)$ and we prove their linear independence using the intertwining operators and the simple current operator.

As for generalization of the result, i.e. the construction of the combinatorial basis for $W(\Lambda)$ when $\Lambda$ is a level two fundamental weight in cases $\ell > 4$, the main problem remains the proof of the linear indpenedence of the spanning set. Namely, it is at the moment unclear how to construct the adequate intertwining operators.

I 