Let (W, S) be a Coxeter system and suppose that w ∈ W is fully commutative (in the sense of Stembridge) and has a reduced expression beginning (respectively, ending) with s ∈ S. If there exists t ∈ S such that s and t do not commute and tw (respectively, wt) is no longer fully commutative, we say that w is left (respectively, right) weak star reducible by s with respect to t. We say that a fully commutative element is non-cancellable if it is irreducible under weak star reductions. In this paper, we classify the noncancellable elements in Coxeter groups of types B and affine C. In a sequel to this paper, the classification of the non-cancellable elements play a pivotal role in inductive arguments used to prove the faithfulness of a diagrammatic representation of a generalized Temperley–Lieb algebra of type affine C.