This root appeared in various notes from the late 1960s as the basis for the particle of uncertainty Q. cé. This particle meant “if” or “maybe” in various contexts, but in notes from 1969 √KE was given the sense “may (be)” and paired with √KWI “suppose”, which was the basis for qui “if” (PE22/158). This can be compared to another paradigm also from 1969 associated with the Ambidexters Sentence where cé meant “if” and it seems the root √keye (with derived verb cíta-) meant “suppose” (VT49/19). Of the two paradigms, I think √KE “maybe”/√KWI “if, suppose” is more useful for the purposes of Neo-Eldarin, but it seems Tolkien’s own thoughts on the subject were in flux.
In notes written in 1967, Tolkien gave what was apparently an inverted form of this root, √ek “it is open”, from which the impersonal Quenya verb ec- “may, can” in the sense “have the opportunity to” is derived (VT49/20). This inverted form is almost certainly related to √KE, as suggested by Patrick Wynne.
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).