  A quadrilateral of factors is an irreducible inclusion of factors $N \subset
M$ with intermediate subfactors $P$ and $Q$ such that $P$ and $Q$ generate $M$
and the intersection of $P$ and $Q$ is $N$. We investigate the structure of a
non-commuting quadrilateral of factors with all the elementary inclusions
$P\subset M$, $Q\subset M$, $N\subset P$, and $N\subset Q$ 2-supertransitive.
In particular we classify such quadrilaterals with the indices of the
elementary subfactors less than or equal to 4. We also compute the angles
between $P$ and $Q$ for quadrilaterals coming from $\alpha$-induction and
asymptotic inclusions.
