  Let K be a field of positive characteristic p and KG the group algebra of a
group G. It is known that if KG is Lie nilpotent then its upper (or lower) Lie
nilpotency index is at most |G'|+1, where |G'| is the order of the commutator
subgroup. The class of groups G for which these indices are maximal or almost
maximal have already been determined. Here we determine G for which upper (or
lower) Lie nilpotency index is the next highest possible.
