  A locally normal subgroup in a topological group is a subgroup whose
normaliser is open. In this paper, we provide a detailed description of the
large-scale structure of closed locally normal subgroups of complete Kac-Moody
groups over finite fields. Combining that description with the main result from
arXiv:2111.07066, we show that under mild assumptions, if the Kac-Moody group
is one-ended (a property that is easily determined from the generalised Cartan
matrix), then it is locally indecomposable, which means that no open subgroup
decomposes as a nontrivial direct product.
