Summary: This work presents further development of the octet formalism established by the authors [Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 458, No. 2022, 1499–1517 (2002; Zbl 1056.74042)] for the classical Kirchhoff anisotropic plate theory. The structure of the fundamental elastic plate matrix is fully explored and the explicit expression is provided. The matrices \(N_{2}\) and \(N_{3}\) are proved to be positive semi-definite. Thus, \(H\) and \(L\) are positive definite. Further studies are concerned with a rotated coordinate system. The transform relation between the eigenvectors in the original and the rotated coordinate system is given. The fundamental elastic plate matrix associated with the eigenrelation referring to the dual coordinate systems, \(N(\theta)\), is studied. The major properties that hold in the Stroh sextic formalism for generalized plane strain problems are also valid in the octet formalism for thin plate bending problems. In particular, we generalize a property in Stroh’s formalism for any non-semisimple matrix \(N(\theta)\). We show a new property in the octet formalism. The non-semisimple cases of \(N(\theta)\) are discussed. Finally, we make it transparent that the mixed/hybrid formalism of others is precisely one of sixteen permuted forms of the octet formalism.