An Etymological Dictionary of the German Language/Annotated/Beil
Beil (Bavarian Beichl), neuter, ‘hatchet,’ from the equivalent Middle High German bîl, bîhel, Old High German bîhal, bîal, neuter (compare the similar stages in the derivation of Feile from fîhala); compare Middle Low German bîl, ‘axe.’ On account of Old Icelandic bílda, ‘axe,’ Old High German bîhal must probably be trace to bîþl, bîtl (for hl from þl compare Gemahl). Hence there may be a connection with the cognates from bhĭ̄d discussed under beißen; as to the meaning, compare especially Latin findo, ‘I split’ (Old Irish biáil, ‘axe,’ is primitively akin). On the other hand, it is, of course, not impossible that Old High German bîhal may be connected with Bicke.