I have a question about a procedure, described in section $2.1.5$ of "Quantum bounded symmetric domains". Here the author describes how to introduce an inner product on $U_q(\mathfrak{g})$. Therefore he uses the isomorphism belonging to the triangular decomposition. But why is the inner product defined on the whole $U_q(\mathfrak{g})$ since in the inner product one only use elements from $U_q(\mathfrak{n}^-)$ and $U_q(\mathfrak{h})$. Can one deduce an explicit expression for example in the case of $\mathfrak{g}=\mathfrak{sl}_2$?
Thanks a lot