Determine conjugacy of the tetracode-compatible gradings

Determine whether the tetracode-compatible gradings of the complex Albert algebra at level n=1 are conjugate to the known fine Z_3^3 grading by comparing their homogeneous products.

Background

The paper constructs compatible tetracode gradings of the complex tip product. At n=1, the resulting homogeneous dimensions are 3+8·3, which agree with a coarsening of the known fine Z_33 grading of the complex Albert algebra. Matching dimensions alone does not establish that the two gradings are equivalent under an automorphism, because their homogeneous multiplication structures may differ.

The authors explicitly leave the conjugacy question unresolved and indicate that a comparison of the homogeneous products is required.

References

At $n=1$, the dimensions in eq:tipFibres are $3+8\cdot3$. They agree with a coarsening of the known fine $\Z_33$ grading of the complex Albert algebra . Whether these gradings are conjugate requires a comparison of their homogeneous products.

eq:tipFibres:

dim⁡(Tn)0=8n−5,dim⁡(Tn)c=1+24n−3(c≠0).\dim(T_n)_0=8n-5,\qquad \dim(T_n)_c=1+2^{4n-3}\quad(c\ne0).

— Okubo algebras and Exceptional Periodicity: tetracode gradings, root-lattice obstructions and Clifford cones  (2610.18643 - Marrani et al., 13 Oct 2026) in Section 6, subsection “Blocks and the compatible grading”