= Solution
The <frequency> and longitudinal <wavevector> formulas are exactly the boost of a contravariant <Lorentz four-vector>:
$$
\begin{pmatrix}\omega'/c\\\mathbf k'_\parallel\end{pmatrix}
=\begin{pmatrix}\gamma&-\gamma\beta\\-\gamma\beta&\gamma\end{pmatrix}
\begin{pmatrix}\omega/c\\\mathbf k_\parallel\end{pmatrix},\qquad
\mathbf k'_\perp=\mathbf k_\perp,
$$
where the matrix is understood on the time component and the signed scalar component along the boost direction. Hence the <four-wavevector> obeys
$$
\boxed{K'^\alpha=\Lambda^\alpha{}_{\beta}K^\beta,\qquad K^\alpha=(\omega/c,\mathbf k).}
$$
For the metric signature $(+,-,-,-)$, lowering the position index gives $R_\alpha=(ct,-\mathbf r)$, so
$$
\boxed{R_\alpha K^\alpha=\omega t-\mathbf k\cdot\mathbf r.}
$$
It is a dimensionless <Lorentz scalar>, equal to minus the <wave phase> used in the stated complex wave. Invariance follows by contracting the mutually inverse position and <wavevector> transformations. Reversing metric signature reverses this displayed scalar's sign, but not its invariance or any observable <phase difference>. In vacuum, $K_\alpha K^\alpha=0$ is an additional null-vector check; the wave remains lightlike in every inertial frame.
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