  Any symmetric closed subset of a finite crystallographic root system must be
a closed subroot system. This is not, in general, true for real affine root
systems. In this paper, we determine when this is true and also give a very
explicit description of symmetric closed subsets of real affine root systems.
At the end, using our results, we study the correspondence between symmetric
closed subsets of real affine root systems and the regular subalgebras
generated by them.
