A short proof that root groups are nilpotent. (Q1882892)

In this note the author gives a surprisingly short, elegant and wonderful proof of the fact that the root groups of a Moufang generalized polygon, and more generally, any Moufang spherical building, are nilpotent. There are not really new applications, but this proof provides a more geometric and direct approach to, or shortcuts in proofs of other, deeper, results like the nonexistence of certain Moufang \(n\)-gons (more precisely, for \(n\neq 3,4,5,6,8\)), and the nilpotency of the unipotent radical of the Borel subgroup of a Moufang BN-pair of rank at least 2.