Commutators associated to a subfactor and its relative commutants

Let N ⊆ M be an inclusion of II1 factors with finite Jones index. Then M = N' ∩ M ⊕ [N, M] as a vector space. Here [N, M] denotes the vector space spanned by the commutators of the form [a, b] where a ∈ N, b ∈ M.