$J$ is a symmetric matrix (built from 6j symbols...it's always knot theory in disguise when I ask :-), $D$ a diagonal matrix, and $B=DJ$. $S$ is a diagonal sign matrix (entries all $+1$ or $-1$). $I$ is the identity matrix. From a bit of diagram juggling, I find the determining equations for $B$ are simply:
Which I then happily solve by brute force (actually some elements of $J$ are known beforehand) but already for a 6*6 matrix this leads to a big mess.
Is there a more "intelligent" method? (I still need to know all elements of $B$...Sidenote: If $S$ contains only one or two $-1$, I already know the closed form for any size of $B$.) Even better, if yes, would it still work if, in a more general version, $S$ is still diagonal but $S^2=I$ no longer holds?