Consider the affine Lie algebra with the Cartan matrix $$ \left(\begin{array}{cc} 2 & -2\\ -2 & 2 \end{array}\right) $$ Let $\omega_{0}$ be the zeroth fundamental weight, $\alpha_1$ the first root, and $\delta$ the imaginary root. We wish to prove that the weights of irreducible highest weight module $L\left(\omega_{0}\right)$ are $$ \left\{ \omega_{0}+n\alpha_{1}-n^{2}\delta-k\delta|n\in\mathbb{Z},k\ge0\right\} (1) $$
Here is the issue: I have easily proved that $$ \left\{ \omega_{0}+n\alpha_{1}-n^{2}\delta|n\in\mathbb{Z}\right\} (2) $$ are weights by applying the Weyl group with basis $s_0,s_1$ to $\omega_0$.
In particular, any $w\in W$ can be expressed in form $\left(s_{0}s_{1}\right)^{n},n\ge0$ or $s_{0}\left(s_{0}s_{1}\right)^{n},n\ge0$. We have that $$ \left(s_{0}s_{1}\right)^{\mathbb N}\omega_0=\left\{ \omega_{0}+n\alpha_{1}-n^{2}\delta|n\in\mathbb{N}\right\} $$ and further $$ s_{0}\left(s_{0}s_{1}\right)^{\mathbb N}\omega_0=\left\{ \omega_{0}+\left(1-n\right)\alpha_{1}-\left(1-n\right)^{2}\delta|n\in\mathbb{N}\right\} $$ which essentially proves (2).
I am obviously missing something very trivial to expand the expression (2) to the required expression (1) for weights, which includes $-k\delta$ as well. I have looked for this bit in textbooks of Kac and Fuchs, but did not find it. I would appreciate a pointer to complete the problem solution.
Is there perhaps an analogue $\sigma$ of a Weil group basis element for imaginary root, which acts on it as $\sigma\left(j\delta\right)=\left(j-1\right)\delta\forall j\in\mathbb{Z}$?