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.
Tolkien (re)introduced the root √IT in 1957 to address a problem he had with the etymology of the name S. Idril. In The Etymologies of the 1930s, he based this name on the root ᴹ√ID “✱desire”, and gave it the form Idhril (Ety/ID). The problem was that in the narratives, he continued to use the form Idril, which could not be derived from √ID. In Notes on Names (NN) he wrote in 1957, he introduced the root √IT to provide a new etymology for Idril, giving the root various meanings such as “(great) enhancement”, “gleaming”, “repeat, multiply” and “glitter, shine, shimmer, twinkle” (PE17/112, 156), along with alternate forms ITH, IS and ƷIT (PE17/156). He seems to have settled on √IT and the last of the listed meanings, since in The Shibboleth of Fëanor written in 1968 Tolkien said her Quenya name was Q. Itarillë based on ita- “to sparkle”, appearing with various other it- words of similar meaning (PM/346, 348 and 363 note #42).
The root √IT was basically a reversion to the original basis for the name G. Idril, which in the 1910s seems to have been based on a (hypothetical) root ✱ᴱ√ITI meaning something like “precious”. For the effects of these revisions on other roots, see the discussion in the entry for √IR “desire”.