allegiance (n.)

"put down, quiet, assuage, pacify," Middle English alegen, from Old English alecgan "to put, place, put down; remit, give up, suppress, abolish; diminish, lessen," from a- "down, aside" (see a- (1)) + lecgan "to lay" (see lay (v.)). A common Germanic compound (cognates: Gothic uslagjan "lay down," Old High German irleccan, German erlegen "to bring down").

Early Middle English pronunciations of -y- and -g- were not always distinct, and the word was confused in Middle English with various senses of Romanic-derived alloy (v.) and especially a now-obsolete verb allege "to alleviate, lighten" (from Latin alleviare, from ad "to" + levis "light" in weight; from PIE root *legwh- "not heavy, having little weight").

Amid the overlapping of meanings that thus arose, there was developed a perplexing network of uses of allay and allege, that belong entirely to no one of the original vbs., but combine the senses of two or more of them. [OED]

Hence the senses "lighten, alleviate; mix, temper, weaken." The confusion with the Latin words probably also accounts for the unetymological double -l-, attested from 17c. Related: Allayed; allaying.

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.).

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."