Etymology
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lather (n.)
Old English leaþr "foam, soap, washing soda," from Proto-Germanic *lauthran (source also of Old Norse lauðr "washing soap, foam"), from PIE *loutro- (source also of Gaulish lautron, Old Irish loathar "bathing tub," Greek louein "to bathe," Latin lavere "to wash"), which is from root *leue- "to wash" + instrumentative suffix *-tro-.

The modern noun might be a 16c. redevelopment from the verb. Meaning "violent perspiration" (especially of horses) is from 1650s; hence the transferred sense "state of agitation" (such as induces sweating), attested from 1839.
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lather (v.)
from a Middle English variant of letheren (v.), from Old English leþran (late West Saxon lyþran) "become covered with (sweat, blood, etc.)," also transitive, from Proto-Germanic *lauthrjan (source also of Old Norse leyðra "to clean, wash;" see lather (n.)). Meaning "to form in froth, produce suds or foam" is from c. 1600. Related: Lathered; lathering.
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*leue- 
*leuə-, Proto-Indo-European root meaning "to wash."

It forms all or part of: ablution; alluvium; deluge; dilute; elution; lather; latrine; launder; lautitious; lavage; lavation; lavatory; lave; lavish; lotion; lye.

It is the hypothetical source of/evidence for its existence is provided by: Greek louein "to wash, bathe;" Latin lavare "to wash," luere "to wash;" Old Irish loathar "basin," Breton laouer "trough;" Old English leaþor "lather," læg "lye."
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