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All of the affirmative singular imperatives of regular verbs are formed by dropping the -re from the second principal part.
Why is it that, in the 3rd conjugation only, the affirmative plural imperative drops the -ere from the second principal part and then adds -ite instead of just dropping the -re and adding -te like in the other conjugations?

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This has to do with the verbal conjugation system of Proto-Indo-European (PIE), the ancestor language of Latin. The Latin third conjugation is largely descended from so-called "thematic" PIE verbs. Thematic verbs are those which have a vowel, either e or o, between the verb root and the ending; this is known as the "thematic vowel". In this case, the vowel was e, so the 2pl. imperative would have ended in -e-te. This e later changed into Latin i by the regular sound change of "vowel reduction", in which basically all short vowels in non-initial open syllables changed into e.

It's not only the i of -ite that is descended from the PIE thematic vowel, but also the i that shows up elsewhere in the third conjugation present forms: ag-i-s, ag-i-t, etc.

(The other three conjugations, too, mostly continue PIE thematic verbs, but in those conjugations the thematic vowel eventually merged into the long vowel of the verb stem.)

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I believe this is due to vowel reduction. Latin vowel reduction is complicated and there seem to be various processes involved; however, a common pattern is vowels turning to /i/.

Syllable and Segment in Latin, by Ranjan Sen (2015), describes a process called "Archaic Latin vowel reduction" whereby short vowels in internal (not initial or final) open syllables were generally all merged to a reduced vowel which became /i/ in Classical Latin. In internal open syllables before /r/, however, the reduced vowel became /e/ instead. This explains the difference between -ite and -ere.

I don't know what the original vowel was in third-conjugation verbs, but it seems like we would get this pattern no matter what it was. It seems to come from the Proto-Indo-European "thematic vowel" (for example, see this page from Allen and Greenough) which apparently could be -e- or -o-. I don't understand the details of its distribution; however, the -u- that appears in Latin verbs of the third declension before nasals in closed syllables (e.g. in third-person plural -unt) seems to descend from PIE -o-.

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