= Solution
The <Einstein's principle of relativity> says that the laws of physics take the same form in every <inertial frame>: no <inertial frame> is physically privileged, and uniform absolute motion cannot be detected by an internal experiment.
Assume $|u|<c$. Direct expansion of the transformed coordinates gives
$$
\begin{aligned}
x'^2-c^2t'^2
&=\gamma^2\left[(x-ut)^2-\left(ct-\frac{u}{c}x\right)^2\right]\\
&=\gamma^2(1-u^2/c^2)(x^2-c^2t^2)=x^2-c^2t^2.
\end{aligned}
$$
Since $y'=y$ and $z'=z$, the full <Minkowski interval> is invariant. Points of the outgoing light shell therefore satisfy
$$
\boxed{x'^2+y'^2+z'^2=c^2t'^2.}
$$
At each fixed $t'>0$ this is a sphere of radius $ct'$. The future branch remains future-directed: for a shell point with $t>0$, $|x|\le ct$ gives $t'=\gamma(t-ux/c^2)>0$.
A <Lorentz transformation> is a linear coordinate transformation preserving the <Minkowski metric>, equivalently $\Lambda^T\eta\Lambda=\eta$. The calculation establishes this property for all events, not only points on the light cone. The displayed map has <determinant> one and preserves <time orientation>, so it is the proper orthochronous boost along the $x$ direction.
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