See also https://familysearch.org/pal:/MM9.1.1/KCRC-662
From the commentary on Gaius Iulius Aiax on OSCAR (regsitered April 2009): "The Greek name <Αἶαξ> (transliterated <Aiax>) appears twice in the Lexicon of Greek Personal Names search at http://www.lgpn.ox.ac.uk/database/lgpn.php . Thus, it appears that the name was occasionally used by persons other than legendary characters. It would not be unusual for a man with the Greek name <Aiax> to use this name as a cognomen if he gained Roman citizenship."