Achiral words

A word $w$ in a free group is {\em achiral} if for every group $G,$ $G_w=G_{w^{-1}},$ where $G_w$ is the image of the word map $w$ on $G\mathrm{.}$ We will give few classes of examples of achiral words. Cocke and Ho asked whether Engel words are achiral or not. We will prove that it is enough to apply Whitehead’s algorithm to check the same.