Derivation property in characteristic 3 when (a,b)=1 and b0≠0

Determine whether the commutator of left multiplication operators [L_a, L_b] is a derivation on the commutative non-associative algebra A with a Frobenius form when a and b are axes satisfying (a,b)=1, the A_0(a)-component b0 of b is nonzero, and the base field F has characteristic 3. Concretely, decide if the conclusion that [L_a, L_b] is a derivation—proved for characteristic not equal to 3 under b0≠0—extends to characteristic 3 under the same hypotheses.

Background

The paper studies a commutative non-associative algebra A over a field F of characteristic not 2 equipped with a Frobenius form, focusing on the structure determined by two axes a and b. A central theme is whether the commutator [L_a, L_b] of the left multiplication operators L_x: y ↦ xy acts as a derivation.

For the special case (a,b)=1, the authors prove that b0∈R, and if b0=0 then [L_a,L_b] is a derivation. Moreover, when b0≠0 and char(F)≠3, they establish that b{}∈R and [L_a,L_b] is a derivation. However, the case char(F)=3 with (a,b)=1 and b0≠0 is not settled; while b0 and b{} are known to lie in the radical R, the derivation property of [L_a,L_b] remains unresolved.