  Further progress in hadron spectroscopy necessitates the phenomenological
description of three particle reactions. We consider the isobar approximation,
where the connected part of the $\mathbf{3}\to\mathbf{3}$ amplitude is first
expressed as a sum over initial and final pairs, and then expanded into a
truncated partial wave series. The resulting unitarity equation is
automatically fulfilled by the $B$-matrix solution, which is an integral
equation for the partial wave amplitudes, analogous to the $K$-matrix
parameterization used to describe $\mathbf{2}\to\mathbf{2}$ amplitudes. We
study the one particle exchange and how its analytic structure impacts
rescattering solutions such as the triangle diagram. The analytic structure is
compared to other parameterizations discussed in the literature. We briefly
discuss the analogies with recent formalisms for extracting
$\mathbf{3}\to\mathbf{3}$ scattering amplitudes in lattice QCD.
