Solution
= Solution
The exterior derivatives of the coframe are
$$
de^0=\sqrt2,e^1\wedge e^2,
\qquad de^1=0,
\qquad de^2=e^1\wedge e^2.
$$
Insert the proposed connection into <Cartan's first structure equation> and use $\omega_{\mu\nu}=-\omega_{\nu\mu}$. The three equations give
$$
B-A=\sqrt2,
\qquad C=-A,
\qquad C=B,
\qquad D=-1.
$$
Thus
$$
\boxed{A=-\frac1{\sqrt2},\qquad B=C=\frac1{\sqrt2},\qquad D=-1}.
$$