The authors determine for precisely which Lie algebras \(L\) over fields of characteristic \(p>0\) the restricted universal enveloping algebra \(u(L)\) is Lie solvable (for \(p>2\)), Lie nilpotent, or satisfies the Engel condition. One of the consequences of these results is that Lie solvability is equivalent to Lie nilpotence in graded group rings of odd characteristic.