**Direct proof** (*to be completed*) generalizing this argument of groups theory :

Let $(N_i \subset M_i)$ be irreducible maximal subfactors.

Let $P$ an intermediate subfactor: $N_1 \otimes N_2 \subset P \subset M_1 \otimes M_2$.

Let $P^1 = \{ x_1 \in M_1 \text{ such that } \exists x_2 \in M_2 \text{ with } x_1 \otimes x_2 \in P \}''$,

and $P_1 = \{ x_1 \in M_1 \text{ such that } x_1 \otimes N_2 \subset P \}''$

Idem, we define $P^2$ and $P_2$.

($\star$) *To be proved*: $(P_i \subset P^i)$ is depth $2$.

Then $(P_i \subset P^i) \simeq (R \subset R \rtimes \mathbb{A}_i)$ with $ \mathbb{A}_i$ a Kac algebra.

Let a $W^*$-isomorphism $\psi_i : R \rtimes \mathbb{A}_i \to P^i $ with $\psi_i(R) = P_i $.

Let $\phi : \mathbb{A}_1 \to \mathbb{A}_2$ with $\phi(a_1) = a_2$ such that $\psi_1(a_1) \otimes \psi_2(a_2) \in P$

($\star$) *To be proved*: $\phi$ is a well-defined isomorphism of Kac algebras.

Then $(P_1 \subset P^1) \simeq (P_2 \subset P^2)$.

Now $N_i \subset P_i \subset P^i \subset M_i$, so by maximality: $P_i, P^i \in \{N_i , M_i \}$.

**If** $(N_1 \subset M_1)$ is depth $>2$ then $P_1=P^1=N_1$ or $M_1$, because $(P_1 \subset P^1)$ is depth $2$,

and $P_2=P^2=N_2$ or $M_2$ because $(P_1 \subset P^1) \simeq (P_2 \subset P^2)$.

($\star$) *To be proved*: $P = \{ x_1 \otimes x_2 \in P \text{ such that } x_1 \in M_1 \text{ and } x_2 \in M_2 \}''$

But $x_1 \otimes x_2 \in P$ implies $x_1 \otimes N_2$, $N_1 \otimes x_2 \subset P$, because $P_i=P^i$.

So $P = P_1 \otimes P_2 \in \{N_1 \otimes N_2 , N_1 \otimes M_2 , M_1 \otimes N_2 , M_1 \otimes M_2 \} $

**If** $(N_2 \subset M_2)$ is depth $>2$, idem...

**If** $(N_1 \subset M_1)$ and $ (N_2 \subset M_2)$ are depth $2$, but are not isomorphic, idem...

**Else** $(N_1 \subset M_1) \simeq (N_2 \subset M_2) \simeq (R \subset R \rtimes \mathbb{A})$ with $\mathbb{A}$ a maximal Kac algebra.

Let $a \not\in \mathbb{C}1$ and $\langle a \rangle$ the left coideal generated by $a$, then by maximalilty $\langle a \rangle = \mathbb{A}$.

So $\mathbb{A} \simeq \langle a \otimes a \rangle \not\in \{ \mathbb{C}\otimes \mathbb{C}, \mathbb{C}\otimes \mathbb{A} , \mathbb{A}\otimes \mathbb{C} , \mathbb{A}\otimes \mathbb{A} \} $, i.e. a non-obvious left coideal of $\mathbb{A}\otimes \mathbb{A}$.

The result follows by Galois correspondence. $\square$

**Corollary of the proof** : the lattice of intermediate subfactors of the tensor product of finitely many irreducible subfactors $(N_i \subset M_i)_{i}$ is the direct product of the lattices $\mathcal{L}(N_i \subset M_i)$ of each subfactors if and only if they are pairwise without isomorphic maximal intermediate depth $2$ inclusions.

*Proof* : the result follows by using $(P_i \subset P^i)$ as above and induction.

**Corollary** : the tensor product of finitely many irreducible cyclic subfactors is a cyclic subfactor if and only if they are pairwise without isomorphic maximal intermediate depth $2$ inclusions.

*Proof* : the direct product of **distributive lattices** is also distributive.

**Remark**: In particular, the tensor product of finitely many irreducible maximal subfactors is a cyclic subfactor if and only if they don't contain two isomorphic subfactors of depth $2$.

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