Triples of long root subgroups

Let G = G(Φ, K) be a Chevalley group over a field K of characteristic ≠ 2. In the present paper, we classify the subgroups of G generated by triples of long root subgroups, two of which are opposite, up to conjugacy. For finite fields, this result is contained in papers by B. Cooperstein on the geometry of root subgroups, whereas for SL (n, K) it is proved in a paper by L. Di Martino and the first-named author. All interesting items of our list appear in deep geometric results on abstract root subgroups and quadratic actions by F. Timmesfeld and A. Steinbach, and also by E. Bashkirov. However, for applications to the groups of type E _l, we need a detailed justification of this list, which we could not extract from the published papers. That is why in the present paper, we produce a direct elementary proof based on the reduction to D ₄, where the question is settled by straightforward matrix calculations. Bibliography: 73 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 343, 2007, pp. 54–83.