The root ᴱ√SAKA first appeared in the Qenya Lexicon of the 1910s with derivatives having to do with “search” (QL/81). The root ᴹ√SAK also appeared in The Etymologies from the 1930s with no gloss or derivatives (EtyAC/SAK). However, as pointed out by Lokyt in a Discord chat on 2019-07-26, the words ᴹQ. sak- “to be hurt” from Quendian & Common Eldarin Verbal Structure (EVS1) of the 1940s and Q. sahta “marred” (< ✶saknā?) from notes written in the late 1950s (MR/405) might be related.
The root appeared as √SAK “draw, pull” in notes associated with Quenya prayers from the 1950s as part of an alternate derivation of Q. úsahtië “inducement to do wrong, ✱temptation” (VT43/23); elsewhere this word was given as a derivative of √THAG “press” (VT43/22). The Sindarin cognate úthaes (VT44/30) is hard to explain as a derivative of √SAK, so it is likely that this was only a transient idea, especially since Tolkien indicates that the older form of the verb sahta- “to induce” was †þahta- (VT43/23).
Neo-Eldarin: For Neo-Eldarin writing, it is probably better to ignore this root and assume its later derivatives are from √THAG. For the senses “draw” and “pull” I recommend using the roots √LUK or ᴹ√TUK instead. In place of ᴱ√SAKA “✱search”, it is probably better to use derivatives of √KETH “examine, ✱seek”.
This root appeared a number of times in Tolkien’s writing over the years, but never with the same meaning. In the Qenya Lexicon of the 1910s, unglossed ᴱ√LUKU appeared with derivatives like ᴱQ. lúke “slime” and ᴱQ. lukso “mud” (QL/56). The Gnomish words G. ûg “mud” and G. ûgrin “muddy” in the contemporaneous Gnomish Lexicon may be related, but if so also indicate a shift in the root to ✱ᴱ√UKU (GL/74).
In The Etymologies of the 1930s the root ᴹ√LUK “magic, enchantment” appeared with derivatives like ᴹQ. lúke “enchantment” and N. lhûth, and the root served as the basis for the name Ilk. Lúthien, translated as “Enchantress” (Ety/LUK). However, in later writings the name S. Lúthien was given a new etymology as a feminization of S. lûth “blossom, inflorescence” (PE17/15).
The root √LUK itself reappeared in notes from around 1968 with the gloss “haul, drag”, serving as the basis for Q. lunka “heavy transport wain (wagon)” (PE17/28; VT43/19). This sense of the root might have an earlier origin, since the verb ᴱQ. luk- “pull” in Early Qenya word lists from the 1920s has a similar meaning (PE16/134).
Another set of possibly related forms are Q. lucassemmar, Q. luciemmar, or Q. luhtammar “✱our debts, our trespasses” from Quenya prayers of the 1950s, as suggested by Wynne, Smith, and Hostetter (VT43/19). These appear to be derived from √LUK, but don’t seem to be connected to any of the attested meanings of this root. In any case, these words for “trespasses” were replaced in later versions of the prayers, becoming Q. rohtammar >> Q. úcaremmar (VT43/19).
Neo-Eldarin: For purposes of Neo-Eldarin, √LUK “haul, drag, ✱pull” is probably the most enduring meaning of this otherwise variable root, but the sense ᴹ√LUK “magic, enchantment” is also very popular, being the basis for some of the better known Elvish words for “magic”. I personally retain both senses, with “magic” by way of analogy for “pulling” on the physical world to affect change. There is a similar analogy to “breath” for the “emission of power (of will or desire)” by analogy with breath fogging a cold surface; see the entry on Q. súlë for discussion. However, this interpretation of √LUK is pure invention on my part to justify retaining the 1930s words for magic.