Heintze, Ernst; Groß, Christian.
Finite order automorphisms and real forms of affine Kac-Moody algebras in the smooth and algebraic category. (English summary)
Mem. Amer. Math. Soc. 219 (2012), no. 1030, viii+66 pp. ISBN: 978-0-8218-6918-5

**Edit**: I recommend the above reference for the following reasons.

Through the work of F. Levstein, G. Rousseau, their collaborators and many others the classification of finite order automorphisms and real forms
of affine Kac-Moody algebras has been completed
(in the algebraic case), after 15 years of work. It is said to fill hundreds of pages.

The work above is supposed to present a much simpler approach which gives more complete results; it also works in the smooth category. In fact the interest of the authors in those questions came from differential geometry,
namely in their study of infinite dimensional symmetric spaces originating from involutions of "smooth" affine Kac-Moody algebras.

Instead of using the structure of Kac-Moody algebras, the authors reduce the problems as fast as they can to the finite-dimensional case. They consider
automorphisms of two kinds, according to whether
they preserve or reverse the orientation of loops,
and show that they have a normal form, which allows to describe them in terms of curves of
automorphisms in the underlying finite-dimensional simple Lie algebra.

In the case of involutive automorphisms, they carry the analysis in detail and deduce a complete classification; this turns out to be
an extension of É. Cartan's classification of symmetric spaces.
They also apply their results in the complex case to conjugate linear involutions and obtain the classification of real forms.