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. In our talk we determine G for which upper (or lower) Lie nilpotency index are maximal or almost maximal, or the next highest possible value.