The first candidates for being the "original references" are the papers by Philip Hall A contribution to the theory of groups of prime-power order [Hall1934] and by Ernst Witt Treue Darstellung Liescher Ringe [Witt1937]. However, none of them, indeed, has the identity in question $\star$ in an explicit form. There is not a single formula with three triple commutators in [Hall1934], whereas [Witt1937] only has an identity $\star\star$, formula (18) on p.156, presenting the product of three triple commutators
$$
[[a,b],c]\cdot [[b,c],a] \cdot [[c,a],b]
$$
as a product of ordinary pairwise commutators (and their conjugates) of $a,b,c$, which is still quite different from $\star$.
There are two expositions coauthored by Wilhelm Magnus, a direct participant in these events. It is Magnus who did the Zbl. Math. review of [Hall1934] (by the way, its JFM review was written by the very same "HNN" Bernhard Neumann whose 1988 paper I quoted in my question; Neumann also reviewed Lazard's paper that I mention below), and then in 1937 he wrote Über Beziehungen zwischen höheren Kommutatoren [Magnus1937] directly inspired by [Hall1934], and which in turn served as an inspiration for [Witt1937] (as explcitly acknowledged in both [Magnus1937] and [Witt1937] published in the same issue of Crelle's).
Theorem 5.1 (called "Witt - Hall" identities", notice the order!) of Combinatorial group theory by Magnus - Karrass - Solitar contains two "elementary" identities involving pairwise commutators (formulas (2.15) and (2.16) of [Hall1934]) together with $\star$ and $\star\star$ (no link between them is mentioned); $\star\star$ is unequivocally atributed to [Witt1937], whereas $\star$ together with the aforementioned two elementary identities are collectively attributed to Hall in a somewhat vague form "see [Hall1934] and [Lazard1954, p.107]" with a reference to Lazard's Sur les groupes nilpotents et les anneaux de Lie [Lazard1954], where one finds an attribution to Hall without any further details whatsoever. However, it is at the very beginning of the introduction, on the first page of the paper, that Lazard also prominently mentions $\star$ referring to it as a "remarquable identité «jacobienne»" and stating that it had been communicated to him by Lev Kaluzhnin (French spelling: Kaloujnine). In the acknowledgement on p. 105 Lazard very warmly thanks Kaluzhnin:
Je tiens à remercier M. L. Kaloujnine pour toute l'aide qu'il m'a apportée, tant au cours de nombreuses conversations personnelles qu'à l'occasion du Séminaire de théorie des groupes qu'il a dirigé à Paris en 1950. Qu'il trouve ici
l'expression de mon amicale reconnaissance.
It is highly unlikely that any communication between Kaluzhnin and Lazard could happen after 1951, the year when Kaluzhnin moved from France to East Germany (and then further to the Soviet Union).
On p.145 of The history of combinatorial group theory by Chandler - Magnus, when discussing various commutator identities, they begin with the aforementioned "elementary" identities, and then say:
These relations were supplemented later by Hall (unpublished) with the identity [... $\star$] which is actually a symmetric form of an identity [$\star\star$] discovered earlier by Witt [1937].
It agrees with what John G. Thompson wrote in Hall's obituary:
Still later [than [Hall1934]] Hall discovered an exact analogue for groups of the Jacobi identity for Lie algebras.
Thus, one can reconstruct the history of the "Hall - Witt" identity $\star$ in the following way:
- 1934 - it does not appear in [Hall1934];
- 1937 - a less symmetric Jacobi type identity $\star\star$ appears in [Witt1937];
- between 1937 and 1950 - Hall (unpublished) finally formulates $\star$.
Philip Hall's apparently still uncatalogued archive might shed some light on how $\star$ became known to the mathematical community.
