O.E.losian “be lost, perish,” from los “destruction, loss,” from P.Gmc. *lausa (cf. O.N. los “the breaking up of an army”), from PIE base *leu– “to loosen, divide, cut apart, untie, separate” (cf. Skt. lunati “cuts, cuts off,” lavitram “sickle;” Gk. lyein “to loosen, untie, slacken,” lysus “a loosening;” L. luere “to loose, release, atone for”). Replaced related leosan (a class II strong verb whose pp. loren survives in forlorn and love-lorn), from P.Gmc. *leusanan (cf. O.H.G. virliosan, Ger. verlieren, O.Fris. urliasa, Goth. fraliusan “to lose”). Transitive sense of “to part with accidentally” is from c.1205. Meaning “to be defeated” (in a game, etc.) is from c.1533. To lose (one’s) mind “become insane” is attested from c.1500. To lose out “fail” is 1858, Amer.Eng.

☛ Online Etymology Dictionary

See the Proto-Indo-European (PIE) etymon adapted from Pokorny: 2. leu- ‘to free, cut off, separate’

From An Etymological Dictionary of the German Language by Friedrich Kluge, trans. John Francis Davis, 1891, p. 220:

los, adjective, ‘loose, released,’ from Middle High German lôs, ‘free, unimpeded, bare, plundered, released, wanton, not solid, frivolous’; corresponding to Gothic laus, ‘empty, invalid, vain,’ Old Icelandic ‘loose, free, unimpeded,’ Anglo-Saxon leás, ‘loose, false, deceitful’ (to this is allied English leas, ‘lie’ and English -less, only as the second part of a compound; English loose is borrowed from Scandinavian), Dutch los, Old Saxon lôs; the adjective form lausa-, common to Teutonic, is from the root lus, ‘to be loose,’ discussed under verlieren. From the Teutonic adjective is derived Spanish lozano, ‘merry, cheerful.’ See lösen.

From Volume 2 of An Etymological Dictionary of Modern English by Ernest Weekley (New York: Dover), 1967, p. 863 (Google Books preview):

lose. AS. losian, to be lost. Orig. intrans., mod. sense coming from association with cogn. leese, AS. leosan (in compds.), which it has supplanted. Leese occurs in A V. (I Kings xviii. 5) where mod. editions have lose ; cf. Du. verliezen, Ger. verlieren, Goth. fraliusan. With leese, lorn, cf. Ger. verlieren (OHG. firliosan), verlorenwas, were.

Below from A Dictionary of English Etymology by Hensleigh Wedgwood, Second Edition, London: Trübner & Co., 1872, p. 397 (the screenshot follows):

Loose. Slack. Du. los, loose,slack, free; Goth. laus, loose, empty, void, of none effect; laus vairthan, to come to nothing; laus as a termination,—less; akranalaus, fruitless; andelaus, endless; lausquithrs, empty-bellied, fasting; lausavaurds, an idle talker; lausjan, to loose, separate, make void.

And finally from Calvert Watkins’s The American Heritage Dictionary of Indo-European Roots, Boston: Houghton Mifflin Company, 1985, pp. 36-37: