On a simple representation of a simple Lie algebra, there is a unique bilinear form for which the actions of $E_i$ and $F_i$ are biadjoint, and some fixed highest weight vector has $\langle v_h,v_h\rangle=1$.

The representation has a distinguished collection of vectors $F_{i_1}\cdots F_{i_n}v_h$ for all sequences $\mathbf{i}$. One can calculate any inner product $\langle F_{i_1}\cdots F_{i_n}v_h, F_{j_1}\cdots F_{j_n}v_h\rangle$, by simply moving the $F_j$'s to become $E_j$'s on the other side, and commuting them past the $F_i$'s. This is not hard to do computationally, but the formulas one gets are not positive, which is annoying for my purposes.

Does anyone know of positive formulae for these inner products? What about their $q$-analogues for quantum groups?