Chirality and non-real elements in $G_2(q)$

In this article, we determine the non-real elements–the ones that are not conjugate to their inverses–in the group $G=G_2(q)$ when $char(F_q)\mathrm{\ne }2,3$. We use this to show that this group is chiral; that is, there is a word w such that $w(G)\mathrm{\ne }w(G{)}^{-1}$. We also show that most classical finite simple groups are achiral