Applications of braided endomorphisms from conformal inclusions

We give three applications of general theory about braided endomorphisms from conformal inclusions developed previously by us. The first is an example of subfactors associated with conformal inclusion whose dual fusion ring is non-commutative. In the second application we show that the Kac-Wakimoto hypothesis about certain relations between branching rules and S-matrices, which has existed for almost a decade, is not true in at least three examples. Finally we show that the fusion rings of subfactors associated with the conformal inclusions $SU(n)_{(n+2)}\subset SU(n(n+1)/2)$ and $SU(n+2)_n\subset SU((n+1)(n+2)/2)$ are canonically isomorphic using a version of level-rank duality.