The geometry of conjugation in Euclidean isometry groups

We describe the geometry of conjugation within any split subgroup $H$ of the full isometry group $G$ of $n$-dimensional Euclidean space. We prove that for any $h\in H$, the conjugacy class $[h{]}_H$ of $h$ is described geometrically by the move-set of its linearization, while the set of elements conjugating $h$ to a given $h^{\mathrm{\prime }}\in [h{]}_H$ is described by the the fix-set of its linearization. Examples include all affine Coxeter groups, certain crystallographic groups, and the group $G$ itself.