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In a comment on this answer, TKR brought up an interesting point of grammar I'd never heard of.

Αἴαντε [Ajax-DUAL] is an interesting case. Though readers of Homer since antiquity have interpreted it as "the two Ajaxes", this seems to be a misunderstanding: it originally meant "Ajax and Teucer", an Indo-European "elliptical dual" (where dual of X means "X and someone closely associated with X").

This sounds interesting, but I've never heard of an "elliptical dual" before. How and when was it used in Greek? Clear, unambiguous examples (i.e. where it's clear it doesn't mean "the two X") would be especially appreciated.

I don't think there are any really clear uses in Greek; the elliptical dual is an archaic Indo-European construction which is no longer productive in Greek, and the elliptical interpretation of Αἴαντε (and the parallel Μολίονε) is based on the usage of other IE languages, mainly Sanskrit.

In Sanskrit, however, the usage is pretty clear. Basically, the idea is that you can refer to a pair of people or things that go together by using the dual form of one of the pair. Some examples:

• 
  pitarau "father-DUAL" = "father and mother"


• 
  mātarau "mother-DUAL" = "mother and father"


• 
  Mitrā "Mitra-DUAL" = "Mitra and Vāruna"


• 
  ahanī "day-DUAL" = "day and night"


• 
  dyāyā "heaven-DUAL" = "heaven and earth"

There are similar examples in Avestan and possibly in Tocharian. And even Latin, which has lost the dual, has an example using plural morphology: Castorēs, which means "Castor and Pollux" (not "several people named Castor").

Common also in classical Arabic, e.g.:

al-qamarāni “the two moons” for “sun and moon”

al-ʼabawāni “the two fathers” for “father and mother”

al-furatāni “the two Euphrates” for “Tigris and Euphrates”

al-ʻumarāni “the two ʻUmars” for the first two caliphs, Abū Bakr and ʻUmar.