I'm looking for a few more examples of using Aschbacher (1980)'s “fundamental SL2 subgroups” description of Sylow 2-subgroups of finite groups of Lie type in odd characteristic. I do not yet understand his original paper, nor the (brief) discussion in GLS (1998), but the use in Harada–Lang (2005) was very compelling.

Can anyone recommend other papers or presentations that might make the description more clear?

- Aschbacher, Michael. “A characterization of Chevalley groups over fields of odd order.” Ann. of Math. (2) 106 (1977), no. 2, 353–398. MR498828
- Aschbacher, Michael. “A characterization of Chevalley groups over fields of odd order. II.” Ann. of Math. (2) 106 (1977), no. 3, 399–468. MR498829
- Aschbacher, Michael. “On finite groups of Lie type and odd characteristic.” J. Algebra 66 (1980), no. 2, 400–424. MR593602
- Gorenstein, Daniel; Lyons, Richard; Solomon, Ronald.
*The classification of the finite simple groups.*Number 3. Part I. Chapter A. Almost simple K-groups. Mathematical Surveys and Monographs, 40.3. American Mathematical Society, Providence, RI, 1998. xvi+419 pp. ISBN: 0-8218-0391-3 MR1490581 - Harada, Koichiro; Lang, Mong Lung. “Indecomposable Sylow 2-subgroups of simple groups.” Acta Appl. Math. 85 (2005), no. 1-3, 161–194. MR2128910