We generalise the two-sided construction of examples of pairs of subfactors of the hyperfinite II$_1$ factor $R$ in [E1]---which arise by considering unitary braid representations with certain properties---to multi-sided pairs. We show that the index for the multi-sided pair can be expressed as a power of that for the two-sided pair. This construction can be applied to the natural examples---where the braid representations are obtained in connection with the representation theory of Lie algebras of types $A$, $B$, $C$, $D$. We also compute the (first) relative commutants.