In fact, the full Lorentz group is not indispensable for studying the Wigner rotation. Given that this phenomenon involves only two spatial dimensions, the subgroup SO(2, 1) is sufficient for analyzing the associated problems. Analogous to the Euler parametrization of SO(3), SO(2, 1) can be decomposed into three simple parts, providing a straightforward and intuitive framework for exploring the Wigner rotation problem.