A note on adjoint reality in simple complex Lie algebras

Abstract: Let $G$ be a Lie group with Lie algebra $\mathfrak g$. In the paper "Reality of unipotent elements in simple Lie groups, Bull. Sci. Math., 185, 2023, 103261" by K. Gongopadhyay and C. Maity, an infinitesimal version of the notion of classical reality, namely adjoint reality, has been introduced. An element $X \in \mathfrak g$ is adjoint real if $-X$ belongs to the adjoint orbit of $X$ in $\mathfrak g$. In this paper, we investigate the adjoint real and the strongly adjoint real semisimple elements in complex simple classical Lie algebras. We also prove that every element in a complex symplectic Lie algebra is adjoint real.