A root appearing in Tolkien’s later writings with a variety of glosses: √MAY “make (in artistic sense as ποιήτης [Greek: make, create])” in Quenya Notes (QN) from 1957 (PE17/145, 163), {√MAGA >>} √(A)MAY “suitable, useful, proper, serviceable; right” in Definitive Linguistic Notes (DLN) from 1959 but this note was crossed through (PE17/172), √MAY “excellent, admirable” elsewhere in DLN (PE17/172), again as √MAY “excellent, admirable” in notes contemporaneous to the Quendi and Eldar essay from 1959-60 (PE17/150, 163) and finally as √MAY “beautiful” in notes from the late 1960s (VT47/18).
Two notable derivatives of √MAY were S. mae “well” (PE17/17, 163) and Q. Maia (PE17/163; VT47/18), but elsewhere these two words were derived from the root √MAG “good (useful)” (PE17/16, 162; PE19/46, 62, 75, 94). In the aforementioned notes from the late 1960s, however, Tolkien said “maga was distinct from maʒa and maya” (VT48/18). For purposes of Neo-Eldarin, I would assume the root √MAY was distinct from √MAG, having the meaning “excellent, admirable, beautiful” and by extension the creation of beautiful things such as art, to allow the retention of 1957 words like Q. maitar “artist” (PE17/163).
A possible precursor to this root is unglossed ᴱ√MAẎA from the Qenya Lexicon of the 1910s with derivatives like ᴱQ. mai “too much” and ᴱQ. maira “excessive, strong” (QL/60); perhaps a more extreme version of its later sense “excellent”. The entry also included ᴱQ. mairu “(horse ?); mane, flowing hair”, but Tolkien marked this word with a “?”, perhaps indicating he was unsure it was from this root (QL/60).
A root whose most notable derivatives are Q. alqua, S. alph “swan”. The earliest iteration of this root was ᴱ√ḶKḶ from the Qenya Lexicon of the 1910s (QL/30); the other derivatives of this root from this period have to do with “appearance” such as ᴱQ. ilk- “to seem” (QL/42). By The Etymologies of the 1930s this root took on its later form, ᴹ√ALAK and had the gloss “rushing” with other derivatives like ᴹQ. alako “rush, rushing flight, wild wind”, N. alag “rushing, impetuous” and N. alagos “storm (of wind)” (Ety/ÁLAK). It was also an element in the name of S./N. Ancalagon “Biting Storm”. Given the continued appearance of this name of The Silmarillion (S/252), the 1930s meaning of this root may have survived, but it is hard to be certain since the name was only properly translated in the 1930s.
The 1930s root also had an unaugmented variant ᴹ√LAK with derivatives ᴹQ. (a)larka, N. lhagr “swift, rapid” (Ety/LAK²). Whether this unaugmented variant remained valid is unclear, but there is nothing in Tolkien’s later writing contradicting it either.