A note on irreducible quadrilaterals of $II_1$ factors

Given any finite index quadrilateral $(N, P, Q, M)$ of $II_1$-factors, the notions of interior and exterior angles between $P$ and $Q$ were introduced in \cite{BDLR2017}. We determine the possible values of these angles when the quadrilateral is irreducible and the subfactors $N \subset P$ and $N \subset Q$ are both regular in terms of the cardinalities of the Weyl groups of the intermediate subfactors. For a more general quadruple, an attempt is made to determine the values of angles by deriving expressions for the angles in terms of the common norm of two naturally arising auxiliary operators and the indices of the intermediate subfactors of the quadruple. Finally, certain bounds on angles between $P$ and $Q$ are obtained, which enforce some restrictions on the index of $N \subset Q$ in terms of that of $N \subset P$.