Solution
= Solution
For the displayed frame,
$$
e^0(e_0)=1,
\quad e^0(e_2)=0,
\quad e^1(e_1)=1,
\quad e^2(e_2)=1,
$$
and all other pairings vanish. Hence the proposed one-forms are the dual coframe. Substitution into
$$
g=-(e^0)^2+(e^1)^2+(e^2)^2
$$
reproduces the metric, proving that the frame is an <orthonormal frame> with signature $(-,+,+)$.