Solution
= Solution
A <four-vector> is an object whose components transform between inertial frames by a <Lorentz transformation>,
$$
U'^\mu=\Lambda^\mu{}_\nu U^\nu.
$$
Lorentz transformations are defined by
$$
\Lambda^T\eta\Lambda=\eta,
$$
where $\eta$ is the <Minkowski metric>. Consequently
$$
U'\cdot U'
=(\Lambda U)^T\eta(\Lambda U)
=U^T\Lambda^T\eta\Lambda U
=U^T\eta U
=U\cdot U.
$$
\b[Thus the <Minkowski norm> of a four-vector is the same in every inertial frame.]