A group $G$ is said to be $n$ engel if $[x,[x,\cdots,[x,y]]\cdots ]=1$ where $x$ appears $n$ times, and this holds for all $x,y\in G$.
In 1955, P M. Cohn gave an example of non-nilpotent group which is $n$-Engel. I was trying to see what this group is? But in the google search, it shows only the reference paper by Cohn. In other words, the group described by Cohn in 1955 doesn't seem to be discussed in any book, and also in expositary notes.
Q. 1 Can one suggest some reference for the description of group in title? Also, is it known that for every $n\geq 2$, there is an $n$-Engel non-nilpotent group?
Q.2 If $G$ is a finite $n$-Engel group, does it follow that $G$ should be nilpotent? how?
Here is a classical example which can be found in some group theory textbooks (I guess it's mentioned in Robinson's): for any prime $p$, the wreath product $\mathbb Z / p \, \textrm{wr} \, (\mathbb Z / p) ^{\infty}$ is $p+1$-Engel and not nilpotent. Replacing $\infty$ with $n$ gives you group which is nilpotent of class $n$. AFAIR, Cohn's original construction was somewhat more involved, but metabelian $p$-group as well.
It's easy to show that every finite Engel group is solvable. To prove that, consider smallest counterexample $G$, which should be simple (obvious). By Sylow theorem, it contains two maximal subgroups intersecting nontrivially; take the pair with maximal intersection, say $R = L \cap M$. $R$ strictly contained in $N_G(R) \cap L$ by nilpotence of $L$. So $R$ is not normal, $N_G(R)$ is proper and contained in some maximal group, but intersecting that maximal one with $M$ gives contradiction with maximality. Then, Gruenberg's theorem assures that finitely generated solvable Engel groups are nilpotent. For detailed proof, see K. Gruenberg, Two theorems on Engel groups, 1953 (it's pretty simple observation on structure of Hall sequences in such groups)