Old English læfan “to allow to remain in the same state or condition; to let remain, allow to survive; to have left (of a deceased person, in reference to heirs, etc.); to bequeath (a heritage),” from Proto-Germanic *laibijan (source also of Old Frisian leva “to leave,” Old Saxon farlebid “left over”), causative of *liban“remain” (source of Old English belifan, German bleiben, Gothic bileiban “to remain”), from root *laf- “remnant, what remains,” from PIE *leip- “to stick, adhere;” also “fat” (cognates: Greek lipos “fat;” Old English lifer “liver,” life).
The Germanic root seems to have had only the sense “remain, continue” (which was in Old English as well but has since become obsolete), which also is in Greek lipares “persevering, importunate.” But this usually is regarded as a development from the primary PIE sense of “adhere, be sticky” (compare Lithuanian lipti, Old Church Slavonic lipet “to adhere,” Greek lipos “grease,” Sanskrit rip-/lip- “to smear, adhere to.”
Originally a strong verb (past participle lifen), it early switched to a weak form. Meaning “go away, take one’s departure, depart from; leave behind” (c. 1200) comes from notion of “leave behind” (as in to leave the earth “to die;” to leave the field “retreat”). From c. 1200 as “to stop, cease; give up, relinquish, abstain from having to do with; discontinue, come to an end;” also “to omit, neglect; to abandon, forsake, desert; divorce;” also “allow (someone) to go.”
Leave, Online Etymology Dictionary