Reducibility of commuting varieties of elements of simple Lie algebra

In this paper, we prove that the variety $C_m(L)$ 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 ${\mathfrak{s}\mathfrak{l}}_2$ and $\mathfrak{s}\mathfrak{l}{)}_3$, and all $m\ge 4$. We also prove it is reducible for $C_3(L)$ for $L$ of types $B_k,C_k,E_7,E_8,F_4,G_2$, $k\ge 2$, as well as for $D_l$ for $l\ge 10$. We do this by proving Theorem on Adding Diagonals, that says that if we can find a simple Lie subalgebra $L^{\mathrm{\prime }}$ whose Dynkin diagram is a subdiagram of the Dynkin diagram of $L$, then under mild conditions, from the fact that $C_m(L^{\mathrm{\prime }})$ is reducible, it follows that $C_m(L)$ is also reducible.