  Lie algebra involutions and their fixed-point subalgebras give rise to
symmetric spaces and real forms of complex Lie algebras, and are well-studied
in the context of symmetrizable Kac-Moody algebras. In this paper we study a
generalization. Namely, we introduce the concept of a pseudo-involution, an
automorphism which is only required to act involutively on a stable Cartan
subalgebra, and the concept of a pseudo-fixed-point subalgebra, a natural
substitute for the fixed-point subalgebra. In the symmetrizable Kac-Moody
setting, we give a comprehensive discussion of pseudo-involutions of the second
kind, the associated pseudo-fixed-point subalgebras, restricted root systems
and Weyl groups, in terms of generalizations of Satake diagrams.
