Etymology
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liege (n.)
late 14c., "vassal of a feudal lord," also "a feudal sovereign, a liege-lord," probably from liege (adj.)) or from a noun use of the adjective in Old French or Anglo-French. A fully reciprocal relationship, so the adjective could apply to either party. Old French distinguished them as lige seignur "liege-lord" and home lige "liege-man."
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liege (adj.)

c. 1300, of lords, "entitled to feudal allegiance and service," from Anglo-French lige (late 13c.), Old French lige "liege-lord," noun use of an adjective meaning "free, giving or receiving fidelity" (corresponding to Medieval Latin ligius, legius), a word of uncertain origin. Perhaps from Late Latin laeticus "cultivated by serfs," from laetus "serf, semi-free colonist," which probably is from Proto-Germanic *lethigaz "freed" (source also of Old English læt "half-freedman, serf;" Old High German laz, Old Frisian lethar "freedman;" Middle Dutch ledich "idle, unemployed"), from extended form of  PIE root *‌‌lē- "to let go, slacken." Or the Middle English word might be directly from Old High German leidig "free," on the notion of "free from obligation to service except as vassal to one lord," but this reverses the notion contained in the word.

From late 14c. of vassals, "bound to render feudal allegiance and service." The dual sense of the adjective reflects the reciprocal relationship it describes: protection in exchange for service. Hence, liege-man "a vassal sworn to the service and support of a lord, who in turn is obliged to protect him" (mid-14c.).

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allegiance (n.)
"ties or obligations of a citizen or subject to a government or sovereign," late 14c., formed in English from Anglo-French legaunce "loyalty of a liege-man to his lord," from Old French legeance, from liege (see liege (adj.)). Corrupted in spelling by confusion with the now-obsolete legal term allegeance "alleviation, mitigation" (for which see allay (v.)). General figurative sense of "recognition of claims to respect or duty, observance of obligation" is attested from 1732. French allégeance in this sense is said to be from English.
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*‌‌lē- 

Proto-Indo-European root meaning "to let go, slacken." 

It forms all or part of: alas; allegiance; lassitude; last (adj.) "following all others;" late; latter; lenient; lenitive; lenity; let (v.) "allow;" let (n.) "stoppage, obstruction;" liege.

It is the hypothetical source of/evidence for its existence is provided by: Greek ledein "to be weary;" Latin lenis "mild, gentle, calm," lassus "faint, weary;" Lithuanian lėnas "quiet, tranquil, tame, slow," leisti "to let, to let loose;" Old Church Slavonic lena "lazy," Old English læt "sluggish, slow," lætan "to leave behind."  

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Beguine (n.)
late 15c., from French béguine (13c.), Medieval Latin beguina, "a member of a women's spiritual order professing poverty and self-denial, founded c.1180 in Liege in the Low Countries." They are said to take their name from the surname of Lambert le Bègue "Lambert the Stammerer," a Liege priest who was instrumental in their founding, and it's likely the word was pejorative at first. French bègue is of unknown origin. Related: Beguinage.

The women's order, though sometimes persecuted, generally preserved its good reputation, but it quickly drew imposters who did not; nonetheless it eventually was condemned as heretical. A male order, called Beghards founded communities by the 1220s in imitation of them, but they soon degenerated (compare Old French beguin "(male) Beguin," also "hypocrite") and wandered begging in the guise of religion; they likely were the source of the words beg and beggar, though there is disagreement over whether Beghard produced Middle Dutch beggaert "mendicant" or was produced by it. The male order was condemned by the Church early 14c. and vanished by mid-16c.

Cole Porter's "Begin the Beguine" (1935) refers to a kind of popular dance of West Indian origin, from French colloquial béguin "an infatuation, boyfriend, girlfriend," earlier "child's bonnet," and before that "nun's headdress" (14c.), from Middle Dutch beggaert, ultimately the same word as the above. Compare English biggin "child's cap" (1520s), from the French word.
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