  We show that the principal order ideal below an element w in the Bruhat order
on involutions in a symmetric group is a Boolean lattice if and only if w
avoids the patterns 4321, 45312 and 456123. Similar criteria for signed
permutations are also stated. Involutions with this property are enumerated
with respect to natural statistics. In this context, a bijective correspondence
with certain Motzkin paths is demonstrated.
