The author provides detailed information for the nilpotency class of Lie algebras (over a field \(k\)) satisfying the Engel condition \((\operatorname {ad}b)^n= 0\) for all \(b\), \(n=3,4\). The author shows that for \(n=3\), \(\operatorname {char}k\neq 2,5\), the nilpotency class is \(\leq 4\) (which is the best possible result) and for \(n=4\), \(\operatorname {char}k\neq 2,3,5\), the nilpotency class is \(\leq 7\). Interesting results for the exceptional case \(n=3\), \(\operatorname {char}k= 2,3\) and \(n=4\), \(\operatorname {char}k= 2,3,5\) are also obtained. For \(n=4\) the author successfully uses the technique of superalgebras.