Geodesic
Classification of homogeneous elements of $\mathbf{Z}_2$-graded semisimple Lie algebras
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 2 (1982), pp. 29-34 
L. V. Antonyan. Classification of homogeneous elements of $\mathbf{Z}_2$-graded semisimple Lie algebras. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 2 (1982), pp. 29-34. http://geodesic.mathdoc.fr/item/VMUMM_1982_2_a7/
@article{VMUMM_1982_2_a7,
     author = {L. V. Antonyan},
     title = {Classification of homogeneous elements of $\mathbf{Z}_2$-graded semisimple {Lie} algebras},
     journal = {Vestnik Moskovskogo universiteta. Matematika, mehanika},
     pages = {29--34},
     year = {1982},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMUMM_1982_2_a7/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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 $\mathfrak{g}_1$.