  Bell inequalities are relevant for many problems in quantum information
science, but finding them for many particles is computationally hard. Recently,
a computationally feasible method called cone-projection technique has been
developed to find all optimal Bell inequalities under some constraints, which
may be given by some symmetry or other linear conditions. In this paper we
extend this work in several directions. We use the method to generalize the
I4422 inequality to three particles and a so-called GYNI inequality to four
particles. Additionally, we find Bell inequalities for three particles that
generalize the I3322 inequality and the CHSH inequality at the same time. We
discuss the obtained inequalities in some detail and characterize their
violation in quantum mechanics.
