Further solvable analogues of the Baer–Suzuki theorem and generation of nonsolvable groups

Let G be an almost simple group. We prove that if x∈G has prime order p⩾5, then there exists an involution y such that 〈x,y〉 is not solvable. Also, if x is an involution then there exist three conjugates of x that generate a nonsolvable group, unless x belongs to a short list of exceptions, which are described explicitly. We also prove that if x has order 6 or 9, then there exist two conjugates that generate a nonsolvable group.