√SAN “think, use mind; ponder, consider in thought”
Tolkien used this root and ones like it on-and-off for Elvish “thought” words throughout his life. The first manifestation of this root was in words from the Gnomish Lexicon of the 1910s like G. sana- “can, know how to, have knowledge, craft or skill” and G. santhi “knowledge, experience (in) or skill (in)” (GL/67), which suggests a (hypothetical) early root *ᴱ√SANA “know”, though such a root does not appear in the contemporaneous Qenya Lexicon. In etymological notes associated with the Ósanwe-kenta essay from 1959-60 Tolkien gave:
√SAM mind, think, reflect, be aware. sāma a mind. sanwe an act of thinking, a thought (*sam-we). ósanwe interchange of thought (between 2 samat). sanwekenda thought-inspection, thought-reading. sanwe-menta thought-sending, mental message (VT41/5).
However, in notes associated with óre that were contemporaneous with The Shibboleth of Feanor from 1968, Tolkien gave Q. sanar = “mind, mind, thinker” (VT41/13), and in Late Notes on Verbs from 1969 (LVS) Tolkien gave the root √SAN “think, use mind; ponder, consider (in thought)” with derivative Q. sanwë “(act or process of) thinking, thought” (PE22/158). Thus 1959-60 √SAM “think” >> 1968-69 √SAN “think”, perhaps a restoration of the Gnomish root from the 1910s.
√SAP “[ᴱ√] dig, excavate”
This root first appeared in the Qenya Lexicon of the 1910s as ᴱ√SAPA “dig, excavate” with derivatives like ᴱQ. sapa- “dig” and ᴱQ. sat (sap-) “hole” (QL/82). At some later point Tolkien wrote ÐAPA as a replacement above this root, but that is not reflected in any of its derivatives. Indeed, the contemporaneous Gnomish Lexicon has a base form sab- and derivatives like G. sabli “spade, shovel” and G. saptha- “to dig, to bite into” (GL/67), indicating ᴱ√SAPA not ᴱ√ÐAPA. Later hints of ᴱ√ÐAPA can be seen in the root ᴹ√DAG “dig” from The Etymologies of the 1930s, but this root was deleted (EtyAC/DAG).
The root √SAP reappeared in a list of roots from the Outline of Phonology (OP2) of the early 1950s with the derivative ✶sapnā > Q. samna “delved hole, pit” (PE19/86). This list was rejected but only because Tolkien revised his thinking on the demonstrated phonetic developments, not the roots themselves. Indeed, the appearance of S. -habar (soft-mutated *sabar) “delving” in various late words like S. Anghabar “Iron-delvings” (S/138) and S. Nornhabar “Dwarrowdelf” (WJ/209) indicate Tolkien probably never really abandoned √SAP “dig”.
√SAR “score, incise; [extended sense] write; [ᴹ√] *stone”
ᴹ√SAR appeared as unglossed root in The Etymologies of the 1930s with derivatives like ᴹQ. sar “(small) stone” and N. sarn “stone as a material” (Ety/SAR). Tolkien use of both these words in later writings in names like Q. Elessar “Elfstone” (LotR/375) and S. Sarn Athrad “Ford of Stones” (S/92) indicates the ongoing validatity of this meaning, though in later writings S. sarn also = “pebble, small stone” (RC/327; VT42/11).
However, in the Quendi and Eldar essay of 1959-60, Tolkien gave a different meaning for this root as √SAR “score, incise”, from which it developed the extended sense “write” and became the basis for the name of the first alphabet of the Elves, the Q. sarati (WJ/396). This use ᴱQ. sar- as “write” dates back to Early Qenya Word-lists of the 1920s (PE16/133). Despite the difference in meaning, I think two root could still be essentially the same, with √SAR = “stone” as a noun and √SAR = “score (stone)” as a verb, and from there eventually developing the sense “write” to serve as the basis for sarati. In one place in the Quendi and Eldar Tolkien revised √SAR >> √SYAR, perhaps indicating he considered making √SYAR an etymological variant of √SAR (WJ/419 note #24).
√SAT “space, place; divide, apportion, mark off”
The root √SAT appeared in notes on The Rivers and Beacon-hills of Gondor from the late 1960s glossed “space, place, a limited area naturally or artificially defined” with various derivatives like Q. sati-/S. seidia- “set aside, appropriate to a special purpose or owner”, Q. satya/S. said “private, separate, not common, excluded”, and S. sad “place, spot” (VT42/19-20). The same root appeared in notes on fractions from 1968 with the gloss “divide, apportion”. It was likely also connected to the primitive suffix ✶-stā that was the basis for the Sindarin/Noldorin collective suffix -ath, as well as the Quenya suffix Q. -sta often seen in place names like the regions of Númenor, and apparently meaning “any particular arrangement, grouping or collection [of things]” (VT39/16, 20 note #27).
A possible early precursor is the (hypothetical) root ᴱ√SATA needed to explain words in Gnomish Lexicon of the 1910s such as G. sad- “reck, care, value, esteem, show respect for, consider” and G. sâd “riches, wealth” (GL/66), but the apparent meaning “*value” is far removed from the later meaning of the root √SAT = “space; apportion”. There was, however, another root √SAT(AR) Quenya Notes (QN) from 1957 with the gloss “faithful, trust, loyal, rely, steadfast” and derivatives like Q. astar/S. astor “faith, loyalty” and Q. satar/S. sadron “trusty follower, loyal companion” (PE17/183), the last appearing as an element in the name S. Echad i Sedryn “Camp of the Faithful” (UT/153). This 1957 root is a bit closer in meaning to ᴱ√SATA “*value”, though still not an exact match.
Neo-Eldarin: For purposes of Neo-Eldarin, I would assume √SAT = “space; apportion” and unrelated √SATAR = “faithful, loyal”.
√SATH “pair”
A root mentioned in passing in Common Eldarin: Noun Structure from the early 1950s as √SATHA “pair” (PE21/74). It had no derivatives in that document, but might be connected to ᴱQ. satto “both” from the 1910s (PE14/76-77) and Ad. satta “two” from the 1940s (SD/428). In the 1950s document, Tolkien wrote in the margin “Best get rid of s, th [for duals] ?”, indicating he likely abandoned √SATHA.
ᴹ√SAY “know, understand”
A root in The Etymologies of the 1930s glossed “know, understand” with derivatives ᴹQ. saira “wise” and ᴹQ. sairon “wizard” (Ety/SAY). It is probably connected to later words like Q. alasaila “unwise” (PE22/154) and S. sael “wise” (MR/305) in documents from the 1950s and 60s, as suggested by Carl Hostetter (VT41/18).
ᴱ√SAẎA “hunger”
A root in the Qenya Lexicon of the 1910s appearing as ᴱ√SAẎA “hunger” mostly used in combination with suffixal -kǝ, having derivatives like ᴱQ. saike “hunger” and ᴱQ. saiqa “hungry” (QL/82). The root also had derivatives in the contemporaneous Gnomish Lexicon like G. saig “hungry” and G. saith “hunger” (GL/66). Some last indications of this root appears in Early Noldorin Word-lists from the 1920s in the words {saig >>} ᴱN. haib “hungry” and {sais >>} ᴱN. hais “hunger”, with Qenya cognates ᴱQ. saisa and ᴱQ. saiste respectively (PE13/146-7).
Neo-Eldarin: For purposes of Neo-Eldarin, I think it is worth positing an Neo-Root ᴺ√SAYAK “hunger” to salvage these early words.