Tolkien used √SI as the basis for “near demonstratives” like “here” and “now” from very early in his writings on Elvish. The Qenya Lexicon of the 1910s had two competing roots ᴱ√HYA “this by us” with derivatives like ᴱQ. hyá “here by us” (QL/41) and ᴱ√KI “this by me” with derivative ᴱQ. tyá (< ᴱ✶kı̯-ā) “now” (QL/41, 49). Indications of the latter can be seen words in the contemporaneous Gnomish Lexicon such as G. cî {“now” >>} “here” and G. cîrin “present (place or time), modern” [gloss deleted] (GL/26). However, Tolkien also introduced a new root ᴱ√si(n) “this here by me” with derivatives like G. sî “here” and G. sith “hither” (GL/68). Revisions of Gnomish ci- word glosses indicate Tolkien was vacillating on which forms were temporal and which were spatial.
In The Etymologies Tolkien gave the root ᴹ√SI “this, here, now” with derivatives like ᴹQ. sí or sin “now” and ᴹQ. sinya/N. sein “new” (Ety/SI). The root √SI was mentioned a couple times in Tolkien’s later writings, usually glossed “this” (PE17/67; VT48/25; VT49/18) and in one place with the variant √SIN (PE17/67). This root was not entirely without competition in Tolkien’s later notes, however: in one place he gave primitive ✶khĭn- as the possible basis for Q. hí “here” and S. hí “now” in 1968 notes on demonstratives, though it appeared beside primitive ✶si- forms (VT49/34 note #21).
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.