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?

**EDIT:**  I should note, following Allen's comment: I'm pretty sure that I know a vector space that has the dimension which is this inner product.  I'm trying to show that a surjective map to this vector space is an isomorphism, and do so by finding a spanning set of the domain that has the right cardinality.