A note on adjoint reality in simple complex Lie algebras (Q6601484)

An element of a group is called ``real'' if it is conjugate to its inverse and ``strongly real'' if it is conjugate to its inverse by an involution. Both real and strongly real elements have been extensively studied in group theory (see, for example, [\textit{A. Singh} and \textit{M. Thakur}, Isr. J. Math. 165, 1--27 (2008; Zbl 1152.20043); \textit{A. A. Gal't}, Sib. Math. J. 51, No. 2, 193--198 (2010; Zbl 1211.20039); translation from Sib. Mat. Zh. 51, No. 2, 241--248 (2010); \textit{P. H. Tiep} and \textit{A. E. Zalesski}, J. Group Theory 8, No. 3, 291--315 (2005; Zbl 1076.20033); \textit{A. G. O'Farrell} and \textit{I. Short}, Reversibility in dynamics and group theory. Cambridge: Cambridge University Press (2015; Zbl 1331.37003)] ).\N\NIn the setting of Lie algebras, the concept of adjoint reality was introduced by the authors in a preceeding work [Bull. Sci. Math. 185, Article ID 103261, 29 p. (2023; Zbl 1520.22011)]. They here show, as a consequence of this definition, that every element in a complex symplectic Lie algebra is adjoint-real.