A root meaning “grow, flourish” used in both Quenya and Sindarin, though in Quenya it was influenced by other roots such as √AL(A) “good, blessed”. Its precursors in the earliest versions of Tolkien’s languages seem to be ᴱ√ALA “spread” (QL/29) and a Gnomish-only root ᴱ√cala [kala], unglossed but with derivatives like G. calw “green shoot, sapling, sprout” and G. caltha- “wax, grow, flourish” (GL/25).
The root ᴹ√GALA first appeared in The Etymologies of the 1930s with various glosses, the last being “thrive” (Ety/GALA). The relevant entries went through considerable revision. For example, the meaning of this root was first given as “grow”, but this meaning was rejected and Tolkien said the verb for “grow” was ol- (EtyAC/GAL(AS)). This new verb seems to be based on a variant root ✱ᴹ√GOL, as indicated by ᴹQ. ola- “grow” < ᴹ✶golā- “grow” appearing in the Quenya Verbal System of the 1940s as a replacement for ᴹQ. ala- < ᴹ✶galā- (PE22/113, note #80), though the new root might also have been ᴹ√OL “grow” as indicated earlier in the same document (PE22/103). Since ᴹ√GALA was retained in The Etymologies, most likely the meaning of this root was changed from “grow” to “thrive” in the 1930s and 40s.
It seems ᴹ√OL “grow” survived into the 1950s (NM/84, 119-120), often with the sense “become” as well (PE22/134); see the entry on √OL for discussion. However, the sense “grow” as also restored to √GAL, but limited mainly to plants. For example, Common Eldarin: Verb Structure from the early 1950s had the primitive verbs ✶galā- “grow (of plants)” vs. ✶olā- “become, come into being, turn into (another state)” (PE22/134). This definition of √GAL as “growth” (along with “flourish” and “be healthy”) appeared regularly in Tolkien’s later writings, though it was often limited primarily to plants, especially in its connection to ✶galadā, the basis for tree words (PE17/25, 135, 153).
A Sindarin-only root for “light; shine”, a variant of √KAL of the same meaning. Its most notable derivative is S. galad “radiance, light”, an element in the names Galadriel and Gil-galad. This root did not explicitly appear in the Qenya Lexicon of the 1910s, where the base for Qenya light words was given as ᴱ√KALA “shine golden” (QL/44), but nearly all the Gnomish derivations of this root begin with g-, as in G. gal- “to shine” and G. glarw(ed) “bright, light” (GL/39). Given that [[eq|initial [g] > [k]]] in early Quenya, this makes it very likely the actual early root was ✱ᴱ√GALA, or at the very least a blending of ᴱ√KALA and ᴱ√GALA.
The first explicit appearance of the root ᴹ√GAL “shine” was in The Etymologies of the 1930s (Ety/GAL), where it was given along with an extended root ᴹ√GALAN “bright” (EtyAC/GAL¹). Tolkien considered but rejected having some Quenya derivatives of these roots: ᴹQ. ala “day” and ᴹQ. alan “daytime” (EtyAC/GAL¹), but he seems to have decided that ᴹ√GAL was a Noldorin-only root, as described in the ᴹ√KAL entry from the same set of documents (Ety/KAL).
The root √GAL reappeared in various etymological notes from the 1950s and 60s, where Tolkien again iterated it was not used in Quenya, and was a root specific to Sindarin (PE17/59) and possibly also Nandorin (PE17/50). The root was potentially problematic in the name Gil-galad, however, in that it was not mutated to Gil-’alad. It seems likely that when Tolkien first coined this name, the second element was from N. calad “light” from the root ᴹ√KAL. This root and word survived into Sindarin (UT/65), and Tolkien sometimes still considered it the basis for Gil-galad (PE17/50).
However, at one point he decided the elements in the names Galadriel and Gil-galad were the same; to explain the lack of mutation in Gil-galad, he posited that the root was actually strengthened to √ÑGAL (PE17/59). Galadriel’s name in The Shibboleth of Fëanor from 1968 was based on the root √ÑAL “shine by reflection” (PM/347), so it seems this was the path Tolkien eventually followed. This makes the ultimate status of √GAL “shine” rather unclear, especially since some of its other derivatives like S. uial twilight (PE17/153) were sometimes derived from √ÑAL (PE17/169).