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We report on transport experiments through high-mobility gate-tunable undoped InSb quantum wells (QWs). Due to the elimination of any Si modulation doping, the gate-defined two-dimensional electron gases in the QWs display a significantly increased mobility of 260000cm^{2}/Vs at a rather low density of 2.4×10^{11}cm^{–2}. Using magnetotransport experiments, we characterize spin-orbit interactions by measuring weak antilocalization. Furthermore, by measuring Shubnikov–de Haas oscillations in tilted magnetic fields, we find that the g factor agrees with k·p theory calculations at low magnetic fields but grows with spin polarization and carrier density at high magnetic fields. Additionally, signatures of Ising quantum Hall ferromagnetism are found at filling factor ν=2 for tilt angles where the Landau level energy equals the Zeeman energy. Despite the high mobility, the undoped InSb QWs exhibit no fractional quantum Hall effect up to magnetic fields of 25 T.