Conjugator length of locally compact groups of Euclidean isometries

We consider locally compact subgroups $H$ of the full isometry group $\mathrm{I}\mathrm{s}\mathrm{o}\mathrm{m}({\mathbb{E}}^n)$ of Euclidean $n$-space which respect the splitting into an orthogonal and a translation subgroup. We prove that the conjugator length function of such groups grows linearly. Our theorem applies, in particular, to the Lie group $\mathrm{I}\mathrm{s}\mathrm{o}\mathrm{m}({\mathbb{E}}^n)$ itself but also to affine Coxeter groups and to split crystallographic groups.