$3$-Engel Group Definitions and Properties This question is motivated by the definition of $2$-Engel group. The following two definitions of a $2$-Engel group are equivalent:


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*A group $G$ such that $x$ commutes with $g^{-1}xg$ for all $x, g\in G$;

*A group $G$ such that $[[x,y],y]=1$ for all $x,y\in G$.


I am wondering if there is a similar equivalence for $3$-Engel group. Namely, are the following two definitions equivalent?


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*A group $G$ such that $\left(g^{-1}xg\right)^{-1}x\left(g^{-1}xg\right)$ commutes with $g^{-1}xg$ for all $x, g\in G$;

*A group $G$ such that $[[[x,y],y],y]=1$ for all $x,y\in G$.


If the answer is "No", does one definition imply the other? Is there a name for the first definition?
More generally, if the first definition is replaced by

A group $G$ such that $\left(g^{-1}xg\right)^{-n}x\left(g^{-1}xg\right)^n$ commutes with $g^{-1}xg$ for all $x, g\in G$; 

how does this definition relates to the definition of $n-$Engle group, which is defined as a group such that 
$$[[[x,y],y],\ldots,y]=1$$ for all $x,y\in G$, where $y$ appears $n$ times?
 A: Note that $g^{-1}xg$ commutes with itself and its inverse, hence $g^{-1}xg$ commutes with $(g^{-1}xg)^{-1}x(g^{-1}xg)$ if and only if it commutes with $x$; the "if" direction is clear, and for the converse, you can use the commutator identity
$$[a,bc] = [a,c][a,b]^c$$
or use the fact the centralizer of $g^{-1}xg$ is a subgroup and contains itself (and hence if it contains $(g^{-1}xg)^{-1}x(g^{-1}xg)$ as well, then it contains $x$). 
Similarly, if $x^g$ commutes with $x$ then it commutes with $(x^g)^{-n}x(x^g)^n$, and if $(x^g)^{-n}x(x^g)^n$ lies in the centralizer of $x^g$, then since so does $x^g$, it follows that $x$ commutes with $x^g$.
So both your conditions labeled "1" are equivalent, as is the condition in the displayed property; they are all equivalent to "$G$ is $2$-Engel".

Perhaps a different way of viewing the two characterizations of $2$-Engel that you give is that the first says: "for every $x\in G$, $x$ commutes with its $G$-conjugates".
Viewed that way, you can find a similar expression for $3$-Engel groups:

The following are equivalent:
  1. For all $x,y\in G$, $[[[x,y],y],y]=1$.
  2. For all $g\in G$, $g$ commutes with all its $[G,g]$ conjugates (that is, conjugates of the form $g^h$ with $h\in[G,g]$).

More generally, $G$ is $n$-Engel if and only if every element $g$ commutes with all its $[G,{}_{n-2}g]$-conjugates.
