On solvable \(\mathbb Z_3\)-graded alternative algebras (Q2798881)

There are well known some sufficient conditions for alternative algebras to be solvable: {\parindent=6mm \begin{itemize}\item[1)] For the alternative algebra \(A\) over a field of characteristic not equal \(2\) and \(g\) an automorphism of the algebra \(A\) of second order, if algebra \(A^{g}\) of fixed points is solvable, then the algebra \(A\) is solvable, see [\textit{O. N. Smirnov}, Sib. Math. J. 32, No. 6, (1991; Zbl 0777.17030); translation from Sib. Mat. Zh. 32, No. 6(190), 158--163 (1991)]. \item[2)] For \(A\) an alternative algebra over a field of characteristic zero and \(G \) a finite group of automorphisms of the algebra \(A\), if the algebra of fixed points \(A^{G}\) is solvable, then the algebra \(A\) is solvable, see [\textit{A. P. Semenov}, Sib. Math. J. 32, No. 1, 169--172 (1991; Zbl 0741.17019); translation from Sib. Mat. Zh. 32, No. 1(185), 207--211 (1991)]. NEWLINENEWLINE\end{itemize}} In this paper, the author studies a special case of the solvability problem for alternative algebras: he considers a \(\mathbb{Z}_{3}\)-graded alternative algebra \(A=A_{0}\oplus A_{1}\oplus A_{2}\) and proves, that if the characteristic of the ground field is not \(2,3\) and \(5\) and \(A_{0}\) is solvable, then the algebra \(A\) is solvable.