Let $A=(a_{ij})$ be a generalized Cartan matrix of order $n$ and $D=diag(d_1,\ldots,d_n)$ the diagonal matrix such that $DA$ is symmetric. Let $$E_{ij}=\sum_{r+s=1-a_{ij}} (-1)^r E_i^{(r)} E_j E_i^{(s)},$$ $i,j\in \{1, \ldots, n\}$, $E_i^{(k)} = \frac{E_i^k}{[k]_{q_i}!}$, $q_i = q^{d_i}$.

Let $$ U_+ = \mathbb{C}(q)\langle E_1, \ldots, E_n \rangle/\langle E_{ij}: i \neq j \rangle. $$

Let $A=\left( \begin{matrix} 2 & -1 \\ -1 & 2 \end{matrix} \right)$. How to show that the set $$B = \{E_1^{(a)}E_2^{(b)}E_1^{(c)}, E_2^{(c)}E_1^{(b)}E_2^{(a)}: b \geq a+c\}$$ is a canonical basis for $U_+$? Where does the condition $b \geq a+c$ come from? Thank you very much.