The Moufang Condition and Root Automorphisms for Spherical Buildings of Rank 3

We give direct, geometric constructions for nontrivial root elations for rank $2$ residues of higher rank buildings $\mathrm{\Delta }$ of type ${\mathsf{B}}_{\mathsf{n}},{\mathsf{C}}_{\mathsf{n}}$ and ${\mathsf{H}}_{\mathsf{m}}$ for $n\in \mathbb{N}$ and $m\in \{3,4\}$. We show that we can extend these to the ambient building in the case that $\mathrm{\Delta }$ has type ${\mathsf{B}}_{\mathsf{n}}$ or ${\mathsf{C}}_{\mathsf{n}}$. With that, we obtain a different proof for the fact that buildings of type ${\mathsf{B}}_{\mathsf{n}}$ and ${\mathsf{C}}_{\mathsf{n}}$ are Moufang. This geometric approach enables us to gain more insight into the root groups associated to these buildings and we obtain new results; Namely, that certain root elations generically fix more points than we previously knew and that every root elation in each point residual can be written as an even self-projectivity. Concerning ${\mathsf{H}}_{\mathsf{m}}$, we will be able to see in a novel way why thick, spherical buildings of type ${\mathsf{H}}_{\mathsf{m}}$ cannot exist. Altogther, this provides an alternative proof for the fact that all thick, irreducible, spherical buildings $\mathrm{\Delta }$ of rank 3 have the Moufang property.