A Concrete 128-Dimensional Khanhnion Algebra with Componentwise Internal Multiplication: Structure Constants, Unit, Rotation Equivariance, and a Lie Bracket on the Vector Sector

This paper presents a concrete 128-dimensional example of the Khanhnion scalar–vector extension. Let W = R³² with componentwise multiplication: β(a, b)_r = a_r b_r. This internal product is commutative, associative, and unital. The resulting algebra is A = (R ⊕ R³) ⊗ W, which has real dimension 128. The paper gives an explicit basis, explicit structure constants, and verifies that the algebra is unital and equivariant under proper spatial rotations. A basis is given by H_(A r) = h_A ⊗ f_r, where h_0, h_1, h_2, h_3 are the quaternion basis elements and f_1, ..., f_32 are the standard basis vectors of R³². The identity is 1_A = h_0 ⊗ (1, 1, ..., 1). The multiplication satisfies (h_A ⊗ a) ⊙ (h_B ⊗ b) = (h_A h_B) ⊗ β(a, b). Because componentwise multiplication is associative, the resulting 128-dimensional algebra is associative. In fact, it is isomorphic to a direct product of 32 copies of the quaternion algebra: A ≅ H³². Spatial rotations act on the three imaginary quaternion directions and leave the 32 internal coordinates unchanged. The algebra is therefore equivariant under every proper rotation in SO(3). The commutator of the algebra has no scalar component. Its restriction to the 96-dimensional vector sector is {x, y} = 2Cβ(x, y), a coefficientwise extension of the ordinary three-dimensional cross product. Since all associators of componentwise multiplication vanish, this bracket satisfies the Jacobi identity. Thus this example realizes the sufficient condition proved in Paper 10 and the associator criterion of Paper 11. The algebra is not a division algebra. For example, the scalar embeddings of two distinct standard basis vectors of R³² multiply to zero. Therefore, A has zero divisors. Also, the full 128-dimensional algebra is not a Lie algebra, because its full product is not anticommutative. Only its commutator restricts to a Lie bracket on the vector sector. This example is best understood as a clear baseline model. It confirms the sufficient condition of Paper 10 and the associator criterion of Paper 11, but it does not produce a new nonassociative algebraic structure. No new sphere-packing density bound, proof of the Kepler theorem, physical quantum model, gauge symmetry, conservation law, or computational advantage is established. Author: Nguyen Kim KhanhProject: Khanh MathematicsDate: 31 October 2026Series: Paper 12 AI assistance was used in mathematical formulation, organization, and wording. Independent verification of the computations, definitions, and relationship to prior work is required before publication.