Jacobi Identity and Lie-Type Structures in the 128-Dimensional Khanhnion Scalar–Vector Extension: A Commutator Bracket, Its Vector-Sector Image, and an Explicit Non-Lie Example

This paper studies whether the commutator of the 128-dimensional Khanhnion scalar–vector extension defines a Lie bracket. Paper 7 introduced the algebra A = (R ⊕ R³) ⊗ W, where W is a 32-dimensional internal algebra. Its product is unital and equivariant under proper spatial rotations. For a commutative internal product β, the commutator of two elements has no scalar component and lies in the 96-dimensional vector sector: [A, B]⊙ = (0, 2Cβ(x, y)). This paper defines the associated bracket on the vector sector by {x, y}β = 2Cβ(x, y). The bracket is bilinear and antisymmetric. The central question is whether it satisfies the Jacobi identity. For the concrete internal choice W = S ⊕ S, where S denotes the real sedenion algebra, with componentwise symmetrized multiplication, the answer is negative. An explicit counterexample is given using the elements x = u₁ ⊗ e₁,y = u₂ ⊗ e₁,z = u₁ ⊗ e₂, where u₁, u₂, u₃ are spatial basis vectors and e₁, e₂ are sedenion basis elements. Using e₁ ∘ e₁ = −1,e₁ ∘ e₂ = 0, the cyclic Jacobi sum becomes Jβ(x, y, z) = −2u₂ ⊗ e₂, which is nonzero. Therefore, the commutator bracket does not satisfy the Jacobi identity. The 128-dimensional scalar–vector extension with this internal choice is not a Lie algebra. The paper identifies the obstruction as the nonvanishing associator of the internal product: assocβ(a, b, c)=β(β(a, b), c) − β(a, β(b, c)). If β is associative, all associators vanish. Associativity is stronger than necessary, but the symmetrized-sedenion product does not satisfy the cancellation conditions needed for Jacobi. The result shows that the framework can produce unital, rotation-equivariant nonassociative algebras, but it does not automatically produce Lie-type symmetry structures. 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: 29 October 2026Series: Paper 9 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.