= Solution
Choose signature $(-,+,+,+)$, with $e^0$ timelike. The <orthonormal coframe in spacetime> reconstructs the <Lorentzian metric> as
$$
\boxed{g=-(e^0)^2+(e^1)^2+(e^2)^2+(e^3)^2=-dt^2+A(t-z)^2dx^2+B(t-z)^2dy^2+dz^2.}
$$
Because $A$ and $B$ are nowhere zero on the domain, the coframe is invertible and the metric nondegenerate; either sign of each function is allowed. In <retarded and advanced null coordinates> $u=t-z$, $v=t+z$ the metric is $-du\,dv+A(u)^2dx^2+B(u)^2dy^2$, a diagonal example of <Rosen coordinates for a plane gravitational wave>. Multiplication by $-1$ would give the opposite overall signature convention; all subsequent signs here use the displayed one.
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