We report a detailed study of the magnetization modulus as a function of temperature and applied magnetic field under varying angle in Sr$_{4}$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 down to 0.48 K. The metamagnetism in Sr$_{4}$Ru$_{3}$O$_{10}$ is accompanied by a reduction of the magnetic moment in the plane of rotation and large field-hysteretic behavior. The double anomaly shifts to higher fields by rotating the field from $H,perp ,c$ to $H,parallel,c$. We compare our experimental findings with numerical simulations based on spin reorientation models caused by intrinsic magnetocrystalline anisotropy and Zeeman effect. Crystal anisotropy is able to explain a metamagnetic transition in the ferromagnetic ordered system Sr$_{4}$Ru$_{3}$O$_{10}$, but a Dzyaloshinskii-Moriya term is crucial to account for a reduction of the magnetic moment as discovered in the experiments.
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