Remark. what the models of 2T retain [fgap-001D]
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    The same binary tetrahedral symmetry now has several models. They answer different questions.

    • The concrete model \(T\leq \mathbb {H}^{\times }\) retains quaternion coordinates, multiplication, conjugation, and norm. It is the model for explicit calculations.
    • The internal semidirect-product structure retains the same group \(T\), but also marks the subgroups \(Q,C\leq T\). The facts \[ T=QC,\qquad Q\mathrel {\trianglelefteq }T,\qquad Q\cap C=\{1\} \] say that every element of \(T\) has a unique factorization \(qc\). They expose how the two parts fit inside the concrete group.
    • The external model \(Q\rtimes _{\alpha }C\) replaces a quaternion by a pair \((q,c)\). It retains the factors and records their interaction in the conjugation action \(\alpha (c)(q)=cqc^{-1}\). Its multiplication is \[ (q,c)(q',c') =\bigl (q\alpha (c)(q'),cc'\bigr ). \]
    • An abstract model can specify the binary tetrahedral group without quaternion coordinates. This is convenient when only its group structure is needed, but coordinates and named factors must be restored by additional maps.

    The multiplication map \[ \mu :Q\rtimes _{\alpha }C\longrightarrow T,\qquad (q,c)\longmapsto qc \] is an isomorphism by the internal semidirect-product theorem. Indeed, \[ \mu \bigl ((q,c)(q',c')\bigr ) =q(cq'c^{-1})cc' =qcq'c' =\mu (q,c)\mu (q',c'). \] Thus the external multiplication is not an analogy: it is exactly the concrete multiplication written in factor coordinates. The two carriers are still different, so a theorem or construction passes between them only through an explicit isomorphism. If it refers to the named factors or their action, the transport must also record the corresponding compatibility.

    The internal and external constructions are developed in [fre2023discrete, sec. 4.2.13, pp. 62--63] and [isaev2018theory, sec. 1.4.2, pp. 58--61]. The concrete identification of \(2T\) with the Hurwitz unit group and \(Q_8\rtimes \mathbb {Z}/3\mathbb {Z}\) is given in [voight2021quaternion, sec. 11.2.4, p. 168].