Subgroups of type \(A_1\) containing a fixed unipotent element in an algebraic group (Q1584605)

The main result proved by the authors is the following theorem: Let \(G\) be a simple algebraic group defined over an algebraically closed field of characteristic \(p>0\). Let \(\sigma\) be either a Frobenius morphism of \(G\), or \(\sigma=1\). Let \(u\in G\) be a \(\sigma\)-invariant element of order \(p\). Then \(u\) is contained in a closed \(\sigma\)-invariant subgroup of \(G\) of type \(A_1\), except in the following cases: (i) \(G=G_2\), \(p=3\), and \(u\) is an element of order \(3\) of type \(A_1^{(3)}\), (ii) \(G=G_2\), \(p=3\), and \(\sigma\) is a morphism involving the graph morphism of \(G\), (iii) \(G=B_2\) or \(G=F_4\), \(p=2\), and \(\sigma\) is a morphism involving the graph morphism of \(G\).