  We continue the study of freely braided elements of simply laced Coxeter
groups, which we introduced in a previous work (math.CO/0301104). A known upper
bound for the number of commutation classes of reduced expressions for an
element of a simply laced Coxeter group is shown to be achieved only when the
element is freely braided; this establishes the converse direction of a
previous result. It is also shown that a simply laced Coxeter group has
finitely many freely braided elements if and only if it has finitely many fully
commutative elements.
