Reducibility of commuting varieties of elements of simple Lie algebra

In this paper, we prove that the variety Cₘ (L) of commuting m-tuples of elements of simple Lie algebra L is often reducible. Explicitely, we prove it is reducible for all simple Lie algebra L not isomorphic to sl₂ and sl) ₃, and all m 4. We also prove it is reducible for C₃ (L) for L of types Bₖ, Cₖ, E₇, E₈, F₄, G₂, k 2, as well as for Dₗ for l 10. We do this by proving Theorem on Adding Diagonals, that says that if we can find a simple Lie subalgebra L' whose Dynkin diagram is a subdiagram of the Dynkin diagram of L, then under mild conditions, from the fact that Cₘ (L') is reducible, it follows that Cₘ (L) is also reducible.