Commuting Squares and Relative Entropy for Two Subfactors

We compute Connes-Störmer relative entropy H(M/N) for two subfactors M and N of a type II1 factor without assuming N ⊂ M. If they form a commuting square, then we have H(M/N) = H(M/M ∩ N). If their commutants form a commuting square, then we have H(M/N) = H(M ∨ N/N).