The aim of the paper is to present a new geometric viewpoint for magnetic flux tubes (MFTs) in the three-dimensional Lorentzian space through split quaternion algebra. For this, we prove that an MFT can be completely characterized by a split quaternion product or a homothetic motion. Moreover, we obtain the magnetic field components by using the split quaternionic expression of an MFT. Then, we show that the flux surfaces can be generated through the magnetic vector field components at each point on a generating curve. The Frenet frame is attached to each point of the generating curve so that the magnetic field components are derived in terms of the local coordinate directions. This allows us to determine the magnetic field lines and kinematic equations of magnetic tube surfaces. Furthermore, characterizations of the MFTs through the stretch factor are presented. Then, the analytical solutions of the kinematic equations are obtained and some examples related to the theory are visualized.