*Hói̯(g/g̑)-o-

Based on the semantic reconstruction as ‘what is given’ and the phonological reconstruction as *h_2/3ói̯(K)‑o- (cf. PAnat. reconstruction above), only one root is available in LIV², viz. PIE 2.*h₂ei̯k̑- ‘(?)to acquire’ (LIV²:223 with n. 1).PIE *Hei̯g̑- ‘to move (heavily)’ (LIV²:222f.), PIE 1.*h₂ei̯k̑- ‘to impale’ (LIV²:259) and PIE *Hei̯g̑^h- ‘to desire’ (LIV²:222) are semantically too remote. Another, merely theoretical, possibility is the root underlying Gk. αἴθω ‘to kindle, ignite’, viz. PIE *h₂ei̯d^h- ‘to ignite’ (LIV²:259) if the root-final *‑d^h- may be separated (< *h₂ei̯- + *d^heh₁- vel sim.). In this case, a root PIE *h₂ei̯- ‘to burn’ vel sim. (IEW:11, but cf. LIV²:229 n. 1 s.v. ?2.*h₁ai‑) could be reconstructed and underly Lyc. B xe/i- ‘offering, sacrifice’ < *h₂ói̯‑o- ‘what is burnt’.In this case, an additional inverse (reflexive?) meaning ‘to give’ has to be accepted, which is not unparalleled, but unattested in the case of this root (cf. LIV²:223, IEW:298f.).For a parallel, the Anatolian reflexes of PIE *deh₃- ‘to give’ may be compared (LIV²:105f.): Luw. /la‑⁽ⁱ⁾/ [lemma link] ‘to take’ vs. Lyc. da‑^(di) ‘to give’ [lemma link]; cf. Sasseville 2021a:543 and also Jasanoff 2003a:211.Furthermore, the exact meaning of the plain root cannot be secured independently because the only formation unambiguously reconstructable for PIE is a (preterito‑)perfect with a resultative meaning ‘to possess, own (lawfully?)’.The most likely option is, of course, the one proposed in LIV²:223 (“sich aneignen”), but there are other possibilities like the more passive ‘to get’, as shown by ModE perf. to have got ‘to possess’ (OED online s.v. get [v.] 33.a), cf. Buck 1949a:741f.
Another possibility depends on the reconstruction of CLuw. iya‑⁽ⁱ⁾ ‘to buy’. If this verb is not related to TochB ai- ‘to give (act.); to take (midd.)’, etc., the root underlying the Core IE formations collected by LIV²:229 should be reconstructed as PIE *h₂ei̯- ‘to give’.
Summing up, the underlying root cannot be determined with certainty because, on the one hand, the necessary phonological shape of the semantically unproblematic root PIE *h₂ei̯- ‘to give’ (instead of 1.*h₁ai̯- as per LIV²:229) is not established (cf. below) and, on the other hand, the phonologically less problematic solution involving PIE 2.*h₂ei̯k̑- ‘to acquire’ (LIV²:223) requires semantic clarification.Seebold 1970a:71f. and EWAhd II:982 tentatively suggest that these two roots may ultimately be related, but fail to explain the function of the palatovelar extension.
For the time being, the connection with the root PIE *h₂ei̯- ‘to give’ (cf. below) is tentatively preferred here as a new proposal.

Based on the Lycian data and the possibly related Hittite verb ḫi(n)k-/ḫai(n)k-^(mi) ‘to bestow, offer’ (cf. above), the root under investigation has to be reconstructed with an anlauting laryngeal PIE *h_2/3- and an i-diphthong, i.e. PIE *h_2/3ei̯‑. However, since Hittite attests a root-final velar which would be lost in Lycian in certain positions, the root might have ended in a velar, i.e. PAnat. *Hei̯(g/g̑)- < PIE *h_2/3ei̯(K)‑, cf. above. Alternatively, the Hittite velar could also be an innovation, cf., e.g., Hitt. haššikk-^(mi) (s.v. PAnat. *HEs(‑K)- ‘to quench one’s thirst’). The only phonologically suitable and semantically reasonable PIE root that is listed in LIV² is 2.*h₂ei̯k̑- ‘to acquire’ (LIV²:223 with n. 1). Phonologically, the reconstruction of the anlauting laryngeal as *h₂ is mainly based on TochB aik- ‘to know’ (cf. LIV²:223 n. 1). While the diphthong is clear from Germanic (*aiχ‑) and Tocharian (aik‑), the auslauting voiceless palatovelar is attested by the Indo-Iranian reflexes (Ved. ī́s-, Av. is‑). Therefore, the root is reconstructed as PIE 2.*h₂ei̯k̑- ‘to acquire’.It is homophonous with PIE 1.*h₂ei̯k̑- ‘to impale’ (LIV²:259), which is otherwise unrelated.
Another possible root is usually reconstructed as PIE 1.*h₁ai̯- (as per LIV²:229). However, its phonological shape is highly dependent upon the root assignment of CLuw. iya‑⁽ⁱ⁾ ‘to buy’. If the Luwic verb does not belong to this root, which is proposed s.v., it may instead be reconstructed as PIE *h₂ei̯- ‘to give’.
Since both roots mentioned entail certain problems, the decision made here in favor of PIE *h₂ei̯‑ ‘to give’ is only tentative.The preterito-present *h₂e-h₂ói̯k̑‑/*h₂e-h₂ik̑‑´ ‘to possess’ (< perfect) is the only formation of the root 2.*h₂ei̯k̑- reconstructable for PIE. On the putative present formations, cf. LIV²:223 n. 2a (tacitly also EDIV:158) and Peyrot 2013a:458f. This makes the above-mentioned proposal of Seebold 1970a:71f. and EWAhd II:982 to connect both roots more attractive. Furthermore, the etymological position of Goth. af-aikan ‘to deny, abjure’ and OHG in-eichan ‘to sacrifice’, etc. has still to be determined (cf., e.g., EWAhd II:970–972, GED:2, EDPG:10).

[A.O.]