Linear Independence for $A_1^{(1)}$ by Using $C_{2}^{(1)}$

In the previous paper, the authors proved linear independence of the combinatorial spanning set for standard $C_\ell^{(1)}$-module $L(k\Lambda_0)$ by establishing a connection with the combinatorial basis of Feigin–Stoyanovsky's type subspace $W(k\Lambda_0)$ of $C_{2\ell}^{(1)}$-module $L(k\Lambda_0)$. In this note we extend this argument for $C_{1}^{(1)}\cong A_{1}^{(1)}$ to all standard $A_{1}^{(1)}$-modules $L(\Lambda)$. In the proof we use a coefficient of an intertwining operator of the type $\binom{L(\Lambda_2)}{L(\Lambda_1)\ L(\Lambda_1)}$ for standard $C_{2}^{(1)}$-modules.