Associated to every subset J of nodes of a Dynkin diagram is a standard parabolic subalgebra pJ of the corresponding Lie algebra g, generated by the positive Borel subalgebra of g together with the negative simple generators fj, j ∈ J . Furthermore one has a decomposition of g as a semi-direct product of pJ and the subalgebra n − J generated by the remaining negative simple generators. In the s...

Lie algebra involutions and their fixed-point subalgebras give rise to symmetric spaces real forms of complex algebras, are well-studied in the context symmetrizable Kac-Moody algebras. In this paper we study a generalization. Namely, introduce concept pseudo-involution, an automorphism which is only required act involutively on stable Cartan subalgebra, pseudo-fixed-point natural substitute fo...