Let $\mathfrak{g}$ be some simple Lie algebra, $\alpha_1,\alpha_2,\cdots,\alpha_n$ be its simple roots. Let $U^+_q(\mathfrak{sl}_2)_i$ be the subalgebra of $U_q(\mathfrak{g})$ generated by $E_i,K_i$. Choose some $w$ in the Weyl group and write $w$ in some reduced expression $$w=s_{i_1}s_{i_2}...s_{i_k}$$ My question is, is $U^+_q(\mathfrak{sl}_2)_{i_1}U^+_q(\mathfrak{sl}_2)_{i_2}\cdots U^+_q(\mathfrak{sl}_2)_{i_k}$ independent of the reduced expression chosen? The background of this question is that I want to define quantum double bruhat cells, and the above subalgebra appears naturally in the definition.
