A structural result of irreducible inclusions of type ${III}_\lambda$ factors, $\lambda\in(0,1)$

Given an irreducible inclusion of factors with finite index $N\subset M$, where $M$ is of type ${III}_{\lambda^{1/m}}$, $N$ of type ${III}_{\lambda^{1/n}}$, $0<\lambda<1$, and $m,n$ are relatively prime positive integers, we will prove that if $N\subset M$ satisfies a commuting square condition, then its structure can be characterized by using fixed point algebras and crossed products of automorphisms acting on the middle inclusion of factors associated with $N\subset M$. Relations between $N\subset M$ and a certain $G$-kernel on subfactors are also discussed.