Twisted-cross-product model in characteristic two

Construct, or prove impossible, a linear isomorphism from the quotient Lie algebra \mathfrak{gl}_2/(\mathbb{F}\cdot I) to \mathbb{F}^3 over a field of characteristic 2 that carries the induced bracket to a twisted cross product E(· × ·) for some invertible E; and determine whether the codimension, three-generator, Cohen–Macaulay, radicality, and primary-decomposition theorems for the recursively defined ideals I_k remain valid verbatim in characteristic 2.

Background

The paper assumes characteristic different from 2 because its encoding separates scalar and traceless parts using division by 2. In characteristic 2, scalar matrices are themselves traceless, so the specific decomposition used in the paper breaks down, even though the commutator still descends to the quotient \mathfrak{gl}_2/(\mathbb{F}\cdot I). The paper reports computational evidence that the recursively defined ideals continue to exhibit the same properties in small characteristic-2 examples, but it does not establish either the existence of a compatible twisted-cross-product model or the general validity of all the main theorems in that characteristic.