  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.
