Are the two-side TLJ subfactors maximal? Let $(N \subset M)$ be a unital inclusion of ${\rm II}_1$ factors, with the following principal graph (called two-side TLJ)

Question: Is $(N \subset M)$ a maximal subfactor?
 A: This depends on your definition of maximal.  There's an intermediate von Neumann algebra which is not a factor, but no intermediate subfactors.
I'll translate this question into a question about tensor categories.  The even part of the subfactor is the category of graded vector spaces $\mathrm{Vec}(\mathbb{Z})$ where I'll denote the 1-dimensional vector space in grade $n$ by $V_n$.  The algebra object is just the matrix algebra $\mathrm{End}(V_0 \oplus V_1)$ whose underlying object is $(V_0 \oplus V_1)(V_0 \oplus V_1)^* = V_0^{\oplus 2} \oplus V_1 \oplus V_{-1}$.  Your question is about whether this algebra has any subalgebras.  Since subalgebras are always self-dual as objects, if it contains $V_1$ it would also have to contain $V_{-1}$ and from that it's pretty easy to see it's the whole thing.  So the only way to get a nontrivial proper subalgebra is $V_0^{\oplus 2}$.  This is obviously a subalgebra because $V_0 \otimes V_0 = V_0$ so the product of any two things in the $V_0$'s has to land back in one of the $V_0$'s.  But this algebra has a nontrivial center.  So it corresponds to an intermediate von Neumann algebra but not an intermediate subfactor.
