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Racah matrices and higher $j$-symbols are used in description of braidingproperties of conformal blocks and in construction of knot polynomials.However, in complicated cases the logic is actually inverted: they are muchbetter deduced from these applications than from the basic representationtheory. Following the recent proposal of arXiv:1612.00422, we obtain theexclusive Racah matrix $\bar S$ for the currently-front-line case ofrepresentation $R=[3,1]$ with non-trivial multiplicities, where it is actuallyoperator valued, i.e. depends on the choice of basises in the intertwinerspaces. Effective field theory for arborescent knots in this case possessesgauge invariance, which is not yet properly described and understood. Becauseof this lack of knowledge a big part (about a half) of $\bar S$ needs to bereconstructed from orthogonality conditions. Therefore we discuss the abundanceof symmetric orthogonal matrices, to which $\bar S$ belong, and explain thatdimension of their moduli space is also about a half of that for the ordinaryorthogonal matrices. Thus the knowledge approximately matches the freedom andthis explains why the method can work -- with some limited addition of educatedguesses. A similar calculation for $R=[r,1]$ for $r>3$ should also be doable.

On moduli space of symmetric orthogonal matrices and exclusive Racah matrix $\bar S$ for representation $R=[3,1]$ with multiplicities