Root subsystems of loop extensions

We completely classify the real root subsystems of root systems of loop algebras of Kac-Moody Lie algebras. This classification involves new notions of "admissible subgroups" of the coweight lattice of a root system $Ψ$, and "scaling functions" on $Ψ$. Our results generalise and simplify earlier work on subsystems of real affine root systems.