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 $C_{\mathrm{\ell }}^{(1)}$-module $L(k{\mathrm{\Lambda }}_0)$ by establishing a connection with the combinatorial basis of Feigin-Stoyanovsky’s type subspace $W(k{\mathrm{\Lambda }}_0)$ of $C_{2\mathrm{\ell }}^{(1)}$-module $L(k{\mathrm{\Lambda }}_0)$. In this note we extend this argument for $C_1^{(1)}\cong A_1^{(1)}$ to all standard $A_1^{(1)}$-modules $L(\mathrm{\Lambda })$. In the proof we use a coefficient of an intertwining operator of the type $\left(\genfrac{}{}{0px}{}{L({\mathrm{\Lambda }}_2)}{L({\mathrm{\Lambda }}_1)L({\mathrm{\Lambda }}_1)}\right)$ for standard $C_2^{(1)}$-modules.