Geodesic
$\mathbb Z_3$-orthograded quasimonocomposition algebras with one-dimensional null component
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 2 (2005), pp. 141-144 
A. T. Gainov. $\mathbb Z_3$-orthograded quasimonocomposition algebras with one-dimensional null component. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 2 (2005), pp. 141-144. http://geodesic.mathdoc.fr/item/SEMR_2005_2_a22/
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     title = {$\mathbb Z_3$-orthograded quasimonocomposition algebras with one-dimensional null component},
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Voir la notice de l'article provenant de la source Math-Net.Ru

We consider $\mathbb Z_3$-orthograded nondegenerate quasimonocomposition algebras $A=A_0\oplus A_1\oplus A_2$ such that $\dim A_0=1$ and $A_1A_2=0$. It is proved that all algebras in this class $W$ are solvable of solvability index either two or three. All non bi-isotropic orthogonal nonisomorphic algebras $A$ of $W$ of least dimension, which is equal to $9$, are classified. An infinite series of algebras $C_r$ in $W$ of dimension $\dim C_r=8r+1$ is constructed for every $r\in\mathbb N=\{1,2,\dots\}$. All algebras $C_r$ are solvable of solvability index $3$ and nilpotent of nil-index $5$.

[1] A. Borel, G. D. Mostow, “On semi-simple automorphisms of Lie algebras”, Ann. Math. (2), 61 (1955), 389–405 | DOI | MR | Zbl

[2] V. A. Kreknin, “Razreshimost algebr Li s regulyarnymi avtomorfizmami konechnogo perioda”, DAN SSSR, 150 (1963), 467–469 | MR | Zbl

[3] E. I. Khukhro, “Gruppy i koltsa Li, dopuskayuschie pochti regulyarnyi avtomorfizm prostogo poryadka”, Matem. sb., 181 (1990), 1207–1219 | MR

[4] N. Yu. Makarenko, E. I. Khukhro, “Pochti razreshimost algebr Li s pochti regulyarnymi avtomorfizmami”, Dokl. RAN, 393 (2003), 18–19 | MR

[5] A. T. Gainov, “$\mathbb Z_n$-ortograduirovannye monokompozitsionnye algebry”, Algebra i logika, 41:1 (2002), 57–69 | MR

[6] A. T. Gainov, “Izomorfizmy konechnomernykh nevyrozhdennykh monokompozitsionnykh algebr”, Matem. zametki, 25 (1979), 801–810 | MR