leyp-
“to smear, to stick”smear, adhere, fat
Root for smearing/sticking, yielding English live/life, Greek lipos (fat), German Leib (body).
Discussion
The PIE root *leyp- (to smear, to stick, to adhere — extended to "to remain, to persist") produced one of the most important life-words in the Germanic branch through a semantic development that equates living with sticking: to be alive is to stick around, to persist, to remain adhering to the world.
English life (OE līf, from PGmc *lībą) descends from this root through the semantic chain: "to stick" → "to remain" → "to be alive" → "life." The word live (OE libban/lifian) continues the verbal form. German Leben (life) and leben (to live) confirm the pan-Germanic distribution.
English liver (OE lifer, from PGmc *librō — "the organ that persists/remains" or "the sticky organ" — the liver's characteristic texture) may derive from the same root, though some scholars connect it to the "life" sense instead (the liver as the life-organ, the organ believed to be the seat of life). Either way, the etymological connection between life and liver is genuine.
The English word leave (as in "to leave behind" — OE lǣfan, "to cause to remain") may also be connected: to leave something is to cause it to stick/stay in place. The noun leavings (remnants, what sticks behind) preserves this sense.
Greek lípos (λίπος, "fat, grease" — the sticky substance) and the derivative lipid (a fatty molecule) preserve the original "smear/stick" sense without the Germanic "life" extension. Liposuction ("fat-sucking") and lipophilic ("fat-loving") derive from the Greek form.
The semantic leap from "stick" to "live" is paralleled in other IE branches: Latin vīvere (to live, from a different root *gʷeyh₃-) shows that the concept of living could be expressed through various physical metaphors. The Germanic choice of *leyp- as the life-root preserved a specifically tactile image: life as adhesion, existence as the refusal to unstick from the world.
Notes
Pokorny 670-671. English life, live, liver from the sticky/fatty sense.