= Solution
At each point, apply the indefinite-signature version of the <Gram-Schmidt process> to any local coframe, choosing one timelike and three spacelike one-forms. This produces an <orthonormal coframe in spacetime> $\{\theta^a\}$ satisfying
$$
g=\eta_{ab}\theta^a\otimes\theta^b,
\qquad \eta_{ab}=\operatorname{diag}(-1,1,1,1).
$$
It is unique only up to a local <Lorentz transformation>. The torsion-free <connection 1 forms> are then determined by <Cartan's first structure equation> and <metric compatibility>,
$$
d\theta^a+\omega^a{}_b\wedge\theta^b=0,
\qquad \omega_{ab}=-\omega_{ba}.
$$
<Cartan's second structure equation> gives the <curvature 2-forms>
$$
\mathcal R^a{}_b=d\omega^a{}_b+\omega^a{}_c\wedge\omega^c{}_b
=\frac12R^a{}_{bcd}\theta^c\wedge\theta^d,
$$
and contraction gives the <Ricci tensor> $R_{bd}=R^a{}_{bad}$.
For the displayed metric, take
$$
\theta^0=dt,
\qquad \theta^i=e^{-t}dx^i
\quad(i=1,2,3).
$$
Then $d\theta^0=0$ and $d\theta^i=-\theta^0\wedge\theta^i$. The nonzero connection forms are therefore
$$
\boxed{\omega^i{}_0=-\theta^i,
\qquad \omega^0{}_i=-\theta^i},
$$
with $\omega^i{}_j=0$. A second application of the structure equations gives
$$
\mathcal R^i{}_0=\theta^0\wedge\theta^i,
\qquad
\mathcal R^0{}_i=\theta^0\wedge\theta^i,
\qquad
\mathcal R^i{}_j=\theta^i\wedge\theta^j.
$$
Equivalently,
$$
\boxed{\mathcal R^a{}_b=\theta^a\wedge\theta_b},
$$
so this is a <constant sectional curvature> spacetime with $K=1$. In four dimensions,
$$
R_{ab}=3g_{ab},
\qquad R=12,
\qquad G_{ab}=R_{ab}-\frac12Rg_{ab}=-3g_{ab}.
$$
The <Vacuum Einstein equations> with a <cosmological constant> are $G_{ab}+\Lambda g_{ab}=0$, and hence
$$
\boxed{\Lambda=3}.
$$
The metric is the contracting flat slicing of <de Sitter spacetime> with curvature radius one.
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