Geodesic
The identity with constants in a~Chevalley group of type~$\mathrm F_4$
Algebra i analiz, Tome 21 (2009) no. 5, pp. 196-202.

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N. L. Gordeev proved that a generalized group identity holds in Chevalley groups with multiply laced root systems. It was also shown that a stronger identity is valid for the Chevalley groups of types $\mathrm{B}_l$ and $\mathrm{C}_l$. In the present paper, it is proved that this strong identity is fulfilled in Chevalley groups of type $\mathrm{F}_4$ and fails to be true in Chevalley groups of type $\mathrm{G}_2$. The main result of the paper is the last ingredient in the proof of the claim that the lattice of intermediate subgroups between $G(\mathrm{F}_4,R)$ and $G(\mathrm{F}_4,A)$ is standard for an arbitrary pair of rings $R\subseteq A$ with $2$ invertible. 
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V. V. Nesterov; A. V. Stepanov. The identity with constants in a~Chevalley group of type~$\mathrm F_4$. Algebra i analiz, Tome 21 (2009) no. 5, pp. 196-202. http://geodesic.mathdoc.fr/item/AA_2009_21_5_a8/

[1] Golubchik I. Z., Mikhalëv A. V.,, “Obobschennye gruppovye tozhdestva v klassicheskikh gruppakh”, Zap. nauch. semin. LOMI, 114, 1982, 96–119 | MR | Zbl

[2] Vavilov N. A., Stepanov A. V., “Nadgruppy poluprostykh grupp”, Vestn. Samar. gos. un-ta. Estestv.-nauch. seriya, 2008, no. 3, 51–95 | MR

[3] Burbaki N., Gruppy i algebry Li. Gruppy Kokstera i sistemy Titsa. Gruppy, porozhdennye otrazheniyami. Sistemy kornei, Mir, M., 1972 | MR | Zbl

[4] Steinberg R., Lektsii o gruppakh Shevalle, Mir, M., 1975 | MR | Zbl

[5] Tomanov G. M., “Obobschennye gruppovye tozhdestva v lineinykh gruppakh”, Mat. sb., 123(165):1 (1984), 35–49 | MR | Zbl

[6] Vavilov N. A., “O geometrii dlinnykh kornevykh podgrupp v gruppakh Shevalle”, Vestn. Leningr. un-ta. Mat., mekh., astronom., 1988, no. 1, 8–11 | MR | Zbl

[7] Gordeev N. L., “Freedom in conjugacy classes of simple algebraic groups and identities with constants”, Algebra i analiz, 9:4 (1997), 63–78 | MR | Zbl

[8] Stepanov A., Free product subgroups between Chevalley groups $\mathrm G(\Phi,F)$ and $\mathrm G(\Phi,F[t])$, Eprint , 2007 http://alexei.stepanov.spb.ru/papers/FreeProd.pdf

[9] Stepanov A., Subring subgroups in Chevalley groups with doubly laced root systems, Eprint , 2009 http://alexei.stepanov.spb.ru/papers/positive.pdf