Efficient topological generation in compact Lie groups

We prove that for any compact simple Lie group G and any nonidentity element g of G the subset of h ∈ G, for which g, h topologically generate G, is nonempty and Zariski open in G. A connected compact Lie group G satisfies 1.5-generation property iff it is simple or abelian. Any compact simple Lie group G has a conjugacy class C such that for every nontrivial elements g₁,g₂ of G there exists y ∈ C so that $\overline{\langle {g}_{1},\,y\rangle }=\overline{\langle {g}_{2},\,y\rangle }=G$. In particular, the generating graph of G has diameter 2.