This verb appeared in some rough notes from around 1969 in various past forms such tanc “fixed”, tancher/tanker “they fixed” and tanchen “I fix” (PE23/138 note #12). See the discussion of the Sindarin past tense for more detail on these past forms. The verb ✱tag- is clearly based on the root √TAK “fix, fasten”.
Conceptual Development: The verb G. tag- “fix, make firm, construct” appeared in the Gnomish Lexicon of the 1910s (GL/68), likely based on the early root ᴱ√TAKA “fix, fasten” (QL/88).
Neo-Sindarin: For purposes of Neo-Sindarin, I would use tag- only in the later sense of its root: “to fix, ✱fasten”. For “make firm” I would use [N.] tangad(a)- and for “construct” I would use [ᴱN.] adaba- “build”.
There were two roots in competition for “construction” words in Tolkien’s later writing: √TAM and √TAN. The first appearance of these was ᴱ√TAMA “(beat) smelt, forge” in the Qenya Lexicon of the 1910s with derivatives like ᴱQ. tambe “copper” and ᴱQ. tamin “forge” (QL/88). In the contemporaneous Gnomish Lexicon it had derivatives like G. tăm “copper” and G. tambos “cauldron” (GL/69).
In The Etymologies of the 1930s the early root reappeared as ᴹ√TAM “knock” with derivatives like ᴹQ. tamba-/N. tamma- “knock” and ᴹQ. tambaro/N. tavor “woodpecker, knocker” (Ety/TAM; EtyAC/TAM). However, for “construction” words Tolkien introduced a new root ᴹ√TAN “make, fashion” with derivatives like ᴹQ. tano “craftsman, smith”, ᴹQ. tanwe “craft, thing made, device, construction” and ᴹQ. kentano/N. cennan “potter” (Ety/TAN). The root ᴹ√TAN was the basis for several names: ᴹQ. Martan(o) “Earth-smith”, ᴹQ. Tintánie “Star-maker”, and ᴹQ. Cirdan “Shipbuilder” (Ety/TAN).
The root √TAN appeared several times in Tolkien’s later writings with the glosses “make with tools” (PE17/96) and “construct” (PE17/106). But the names ᴹQ. Martan(o) and ᴹQ. Tintánie became Q. Martamo (LT1A/Talka Marda) and Q. Tintallë (PE21/85; LotR/377), and in notes from 1968 or 1969 Tolkien had {√TAN >>} √TAM “construct” with derivatives like Q. taman/S. tavn “a thing made by handicraft” and Q. tamo “smith” (PE17/107-108). However, in this same note Tolkien said “in Sindarin the base appeared mostly in form √TAN owing to contact with √PAN arrange, set in order”, most likely because he wanted to retain names like S. Círdan.
Neo-Eldarin: For purposes of Neo-Eldarin, I would assume the root √TAM originally had the sense “beat, knock”, from there becoming the basis for “forge” words and then a general root for tool and handicraft words, but with a Sindarin-only variant √TAN as noted above. I would use the root √TAN for “indicate, show”, see that entry for details.