  Let $(W,S)$ be a Coxeter system and suppose that $w \in W$ is fully
commutative (in the sense of Stembridge) and has a reduced expression beginning
(respectively, ending) with $s \in S$. If there exists $t\in S$ 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 $t$. In this paper, we classify the fully commutative
elements in Coxeter groups of types $B$ and affine $C$ that are irreducible
under weak star reductions. In a sequel to this paper, the classification of
the weak star irreducible elements in a Coxeter system of type affine $C$ will
provide the groundwork for inductive arguments used to prove the faithfulness
of a generalized Temperley--Lieb algebra of type affine $C$ by a particular
diagram algebra.
