Inverse of right engel element is a left engel element.

Question

Heineken shows that inverse of a right engel element is a left engel element. For this he showed the commutator identity $[x,_{n+1} g]=^g [^xg^{-1},_n g]$. I am studying the textbook "A course in the theory of groups" by Robinson. I am not able to conclude the claimed result from this commutator identity. Please help me.

Answer 1

$[x,_{n+1}g]=[[x,g],_ng]=[x^{-1}g^{-1}xg,_ng]=[(g^{-1})^xg,_ng]=[[(g^{-1})^xg,g],_{n-1}g]=[[(g^{-1})^x,g]^g,_{n-1}g]=[(g^{-1})^x,_ng]^g$

Answer 2

First I apologize for such question. The answer of the above question, if g is an right engel element then so $^xg$ also. Now by using the given commutator identity we can have the desired result.