  Let K be a field of characteristic 2 and G a nonabelian locally finite
2-group. Let V(KG) be the group of units with augmentation 1 in the group
algebra KG. An explicit list of groups is given, and it is proved that all
involutions in V(KG) commute with each other if and only if G is isomorphic to
one of the groups on this list. In particular, this property depends only on G
and not at all on K.
