Lie Sectors and the Universal Associative Quotient of the Khanhnion Cross-Type Bracket

Paper 16 proves that, for a commutative internal multiplication, the Khanhnion cross-type bracket satisfies Jacobi precisely when the internal multiplication is associative. This note studies two compatible consequences. First, rotation-invariant Lie sectors correspond exactly to associative internal subalgebras. Second, the ideal generated by all internal associators yields a canonical associative quotient and therefore a current Lie algebra. We prove its universal property for product-preserving internal maps and, separately, for arbitrary Lie-valued bracket homomorphisms. In the unital case, the latter result follows by identifying the ideal generated by Jacobi defects. Examples distinguish Lie subalgebras from Lie quotients and show that the universal Lie quotient can collapse to zero. No classification of all Lie subalgebras, computational advantage, or priority claim is made.