Classification of homogeneous elements of $\mathbf{Z}_2$-graded semisimple Lie algebras

Let $\mathfrak{g}=\mathfrak{g}_0\oplus\mathfrak{g}_1$ be a $\mathbf{Z}_2$-graded semisimple Lie algebra over the complex number field. Our purpose is threefold. First, we describe the conjugacy classes of $\mathfrak{g}$ that intersect $\mathfrak{g}_1$. Second, we show that all classes intersect $\mathfrak{g}_1$ if and only if $\mathfrak{g}_1$ contains a Cartan subalgebra of $\mathfrak{g}$. Third, we obtain some necessary and sufficient conditions, under which all classes of nilpotent elements in $\mathfrak{g}$ intersect $\mathfrak{g}_1$.