Unresolved case (a,b)=1/4

Determine the behavior of the identities and structural conclusions obtained in the paper for the unresolved case where the Frobenius inner product of the axes satisfies (a,b)=1/4. In particular, establish whether the identity b_{}(x_0 y_0)=(b_{}x_0)y_0+x_0(b_{}y_0—proved for (a,b)≠1/4—holds when (a,b)=1/4, and clarify how the results about [L_a, L_b] being a derivation extend to this case.

Background

Several key results in the paper, such as Theorem 0012 proving b_{}(x_0y_0)=(b_{}x_0)y_0+x_0(b_{}y_0), explicitly exclude the case (a,b)=1/4. Likewise, the main derivation result for [L_a,L_b] is established for (a,b) not in {0,1,1/4}, with separate treatment for (a,b)=0 and (a,b)=1. The case (a,b)=1/4 remains unaddressed in this work.

The authors state that they have left the case (a,b)=1/4 to future work, indicating that the structural identities and derivation properties in this borderline situation have not been determined.