Let $(N \subset M)$ be a unital inclusion of ${\rm II}\_1$ factors, the basic construction is $N \subset M \subset M\_1 = \langle M , e^M\_N \rangle$.


*Question:* For any intermediate subfactor $N \subset P \subset M\_1$, is it true that $N \subset P \subseteq M$ or $M \subset P \subset M\_1$ (i.e. no extra intermediate)?



Is there no extra intermediate subfactor for the basic construction?
This answer came by a discussion with Keshab Chandra Bakshi.  

Consider the subfactor $(R \subset R \rtimes \mathbb{Z}/2)$, then $R \subset R \rtimes \mathbb{Z}/2 \subset M\_2(R) = R \otimes M\_2(\mathbb{C})$ is the basic construcition, but $(R \subset M\_2(R))$ admits continuously many intermediate subfactors which are conjugate to $R \rtimes \mathbb{Z}/2$ by unitaries in $\mathbb{C} \otimes M\_2(\mathbb{C})$. So the answer is no.


