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Let $G$ be an almost simple group. We prove that if $x \in G$ has prime order$p \ge 5$, then there exists an involution $y$ such that $<x,y>$ is notsolvable. 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 ofexceptions, which are described explicitly. We also prove that if $x$ has order$6$ or $9$, then there exists two conjugates that generate a nonsolvable group.

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