:: The identity in question is not just some identity that happens to hold in a group. It's a rather important one (just like division) to the extent that (like division which taken separately leads to quasigroups) the identity in question leads to semigroups with involution of which are also plentiful examples. I've seen various lame excuses above for the sorry state of this "awesome" article. They usually involve some talking down any concerns and general hand-waving toward some WP:THISORTHAT just like you did. Sayonara.