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 $L(kLambda_0)$ by establishing a connection with the combinatorial basis of Feigin-Stoyanovsky's type subspace $W(kLambda_0)$ of $C_{2ell}^{(1)}$-module $L(kLambda_0)$. 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.