Geodesic
$\mathrm{SL}_2$-factorisations of Chevalley groups
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 22, Tome 394 (2011), pp. 20-32

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Recently Liebeck, Nikolov, and Shalev noticed that finite Chevalley groups admit fundamental $\mathrm{SL}_2$-factorizations of length $5N$, where $N$ is the number of positive roots. From a recent paper by Smolensky, Sury, and Vavilov it follows that elementary Chevalley groups over rings of stable rank 1 admit such factorizations of length $4N$. In the present paper, we establish two further improvements of these results. Over any field the bound here can be improved to $3N$. On the other hand, for $\mathrm{SL}(n,R)$, over a Bezout ring $R$, we further improve the bound to $2N=n^2-n$. 
@article{ZNSL_2011_394_a1,
     author = {N. A. Vavilov and E. I. Kovach},
     title = {$\mathrm{SL}_2$-factorisations of {Chevalley} groups},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {20--32},
     publisher = {mathdoc},
     volume = {394},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2011_394_a1/}
}
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N. A. Vavilov; E. I. Kovach. $\mathrm{SL}_2$-factorisations of Chevalley groups. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 22, Tome 394 (2011), pp. 20-32. http://geodesic.mathdoc.fr/item/ZNSL_2011_394_a1/