Let $��$ be an involution of a complex semisimple Lie algebra $\mathfrak g$ and $\mathfrak g=\mathfrak g_0\oplus\mathfrak g_1$ the related $\mathbb Z_2$-grading. We study relations between nilpotent $G_0$-orbits in $\mathfrak g_0$ and the respective $G$-orbits in $\mathfrak g$. If $e\in\mathfrak g_0$ is nilpotent and $\{e,h,f\}\subset\mathfrak g_0$ is an $\mathfrak{sl}_2$-triple, then the semisimple element $h$ yields a $\mathbb Z$-grading of $\mathfrak g$. Our main tool is the combined $\mathbb Z\times\mathbb Z_2$-grading of $\mathfrak g$, which is called a mixed grading. We prove, in particular, that if $e_��$ is a regular nilpotent element of $\mathfrak g_0$, then the weighted Dynkin diagram of $e_��$, $\mathcal D(e_��)$, has only isolated zeros. It is also shown that if $G{\cdot}e_��\cap\mathfrak g_1\ne\varnothing$, then the Satake diagram of $��$ has only isolated black nodes and these black nodes occur among the zeros of $\mathcal D(e_��)$. Using mixed gradings related to $e_��$, we define an inner involution $\check��$ such that $��$ and $\check��$ commute. Here we prove that the Satake diagrams for both $\check��$ and $��\check��$ have isolated black nodes.

23 pages