We construct a simple algorithm to generate any Clauser-Horne-Shimony-Holt- (CHSH-) type Bell inequality involving a party with two local binary measurements from two CHSH-type inequalities without this party. The algorithm readily generalizes to situations where the additional observer uses three measurement settings. There, each inequality involving the additional party is constructed from three inequalities with this party excluded. With this generalization at hand, we construct and analyze a class of symmetric inequalities for four observers and three experimental settings per observer.