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In this paper, triality refers to the $S_3$-symmetry of the language of quasigroups, which is related to, but distinct from, the notion of triality as the $S_3$-symmetry of the Dynkin diagram $D_4$. The paper investigates a homogeneous method for rendering the linearization of quasigroups (over a commutative ring) naturally invariant under the action of the triality group, on the basis of an appropriate algebra generated by three invertible, non-commuting coefficient variables that is isomorphic to the group algebra of the free group on two generators. The algebra has a natural quotient given by setting the square of each generating variable to be $-1$. The quotient is an algebra of quaternions over the underlying ring, in a way reminiscent of how symmetric groups appear as quotients of braid groups on declaring the generators to be involutions. The corresponding quasigroups (which are described as quaternionic) are characterized by three equivalent pairs of quasigroup identities, permuted by the triality symmetry. The three pairs of identities are logically independent of each other. Totally symmetric quasigroups (such as Steiner triple systems) are quaternionic.