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We report a detailed study of the magnetization modulus as a function oftemperature and applied magnetic field under varying angle inSr$_{4}$Ru$_{3}$O$_{10}$ close to the metamagnetic transition at$H_{c}\backsimeq 2.5\,$T for $H \perp c$. We confirm that the double-feature at$H_{c}$ is robust without further splitting for temperatures below 1.8 K downto 0.48 K. The metamagnetism in Sr$_{4}$Ru$_{3}$O$_{10}$ is accompanied by areduction of the magnetic moment in the plane of rotation and largefield-hysteretic behavior. The double anomaly shifts to higher fields byrotating the field from $H\,\perp \,c$ to $H\,\parallel\,c$. We compare ourexperimental findings with numerical simulations based on spin reorientationmodels caused by intrinsic magnetocrystalline anisotropy and Zeeman effect.Crystal anisotropy is able to explain a metamagnetic transition in theferromagnetic ordered system Sr$_{4}$Ru$_{3}$O$_{10}$, but aDzyaloshinskii-Moriya term is crucial to account for a reduction of themagnetic moment as discovered in the experiments.

Missing magnetism in Sr$_{4}$Ru$_{3}$O$_{10}$: Indication for Antisymmetric Exchange Interaction