Geodesic
The Engel identity and its application
Sbornik. Mathematics, Tome 49 (1984) no. 2, pp. 419-426 
S. P. Mishchenko. The Engel identity and its application. Sbornik. Mathematics, Tome 49 (1984) no. 2, pp. 419-426. http://geodesic.mathdoc.fr/item/SM_1984_49_2_a8/
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Voir la notice de l'article provenant de la source Math-Net.Ru

In the case of characteristic zero, the Engel identity implies nilpotence in the variety generated by simple infinite-dimensional Lie algebras of Cartan type. An analogous result is also true for 2-metabelian Lie algebras (an algebra is called 2-metabelian if every 2-generator subalgebra is metabelian) over a field whose characteristic does not divide $5!\,$, which in this case permits one to prove solvability of the variety of 2-metabelian Lie algebras. Bibliography: 10 titles.

[1] Higgins P. J., “Lie rings satisfying the Engel condition”, Proc. Cambr. Phylos. Soc., 50:1 (1954), 8–15 | DOI | MR | Zbl

[2] Kostrikin A. I., “O lokalnoi nilpotentnosti kolets Li, udovletvoryayuschikh usloviyu Engelya”, DAN SSSR, 118:6 (1958), 1047–1077 | MR

[3] Neiman X., Mnogoobraziya grupp, Mir, M., 1969 | MR

[4] Vaughan-Lee M. R., Varieties of Lie algebras, D. Phyl. Thesis, Oxford, 1970 | MR

[5] Bahturin J. A., Lectures on Lie algebras, Akademie-Verlag, Berlin, 1978 | MR | Zbl

[6] Drenski V. S., “Predstavleniya simmetricheskoi gruppy i mnogoobraziya lineinykh algebr”, Matem. sb., 115 (157) (1981), 98–114 | MR

[7] Klyachko A. A., “Elementy Li v tenzornoi algebre”, Sib. matem. zh., 15:6 (1974), 1296–1304 | Zbl

[8] Mischenko S. P., “Mnogoobraziya algebr Li so slabym rostom posledovatelnosti korazmernostei”, Vestn. MGU, 1982, no. 5, 63–66 | Zbl

[9] Razmyslov Yu. P., “Ob engelevykh algebrakh Li”, Algebra i logika, 10:1 (1971), 33–44 | MR | Zbl

[10] Sheina G. V., “O nekotorykh mnogoobraziyakh lievykh algebr”, Sib. matem. zh., 17:1 (1976), 194–199 | MR | Zbl