 Relative speed from four-velocity contraction

For future-directed unit timelike four-velocities in metric signature $(+---)$, $U_1\cdot U_2/c^2$ is the Lorentz factor of either observer measured in the rest frame of the other. In a common inertial frame it equals ${\gamma }_1{\gamma }_2(1-{\mathbf{\beta }}_1\cdot {\mathbf{\beta }}_2)$. ThereforeExpanding the numerator converts this to the dot-and-cross-product form of relativistic velocity addition. The invariant construction explains why the relative speed is symmetric even when the spatial velocity components belong to different rest frames.