= Solution
An <orthonormal coframe> writes a metric as $g=\eta_{ab}\theta^a\otimes\theta^b$ with constant diagonal signature <matrix> $\eta$. In <Cartan's first structure equation>, solve
$$
d\theta^a+\omega^a{}_b\wedge\theta^b=0,\qquad
\omega_{ab}=-\omega_{ba},\qquad \omega_{ab}=\eta_{ac}\omega^c{}_b.
$$
These are the torsion-free and metric-compatibility conditions and determine the <Levi-Civita connection>. Then <Cartan's second structure equation> gives
$$
\Omega^a{}_b=d\omega^a{}_b+\omega^a{}_c\wedge\omega^c{}_b
=\frac12 R^a{}_{bcd}\theta^c\wedge\theta^d.
$$
Contract $R^a{}_{bad}$ to obtain the <Ricci tensor> and then contract again for the <scalar curvature>. This fixes the curvature convention; reversing the definition of the <Riemann curvature tensor> reverses the resulting curvature signs.
On $z>0$, choose $\theta^0=dt/z$, $\theta^1=dx/z$, $\theta^2=dy/z$, $\theta^3=dz/z$, with $\eta=\operatorname{diag}(-1,1,1,1)$. For $a=0,1,2$,
$$
d\theta^a=\theta^a\wedge\theta^3,\qquad d\theta^3=0.
$$
A torsion-free metric connection therefore has
$$
\omega^a{}_3=-\theta^a,\qquad
\omega^3{}_a=\eta_{aa}\theta^a,\qquad
\omega^a{}_b=0\quad(a,b<3).
$$
For example, $\Omega^a{}_3=-d\theta^a=-\theta^a\wedge\theta^3$, while for $a,b<3$,
$$
\Omega^a{}_b=\omega^a{}_3\wedge\omega^3{}_b
=-\eta_{bb}\theta^a\wedge\theta^b.
$$
Metric antisymmetry supplies the remaining components, giving uniformly
$$
\boxed{\Omega^a{}_b=-\theta^a\wedge\theta_b,\qquad
R_{abcd}=-(\eta_{ac}\eta_{bd}-\eta_{ad}\eta_{bc}).}
$$
Thus the metric has constant <sectional curvature> $-1$. In four dimensions,
$$
\boxed{\operatorname{Ric}_{\mu\nu}=-3g_{\mu\nu},\qquad R=-12.}
$$
It is an <Einstein manifold>, with Einstein constant $-3$ in the stated curvature convention.
These are <Poincare coordinates on anti-de Sitter spacetime> of unit radius. Constant curvature gives the maximal ten-dimensional local <isometry> algebra $\mathfrak{so}(3,2)$. This can also be checked directly: set $u=(t,x,y)$, $\eta_{ab}=\operatorname{diag}(-1,1,1)$ and $u_a=\eta_{ab}u^b$. The ten independent <Killing vector fields> are
$$
P_a=\partial_a,\qquad
M_{ab}=u_a\partial_b-u_b\partial_a,\qquad
D=u^a\partial_a+z\partial_z,\qquad
K_a=2u_aD-(u^bu_b+z^2)\partial_a.
$$
Translations and <Lorentz transformations> leave the numerator and $z$ unchanged. A dilation rescales numerator and denominator equally. For $K_a$, direct differentiation gives $\mathcal L_{K_a}\eta^{(4)}=4u_a\eta^{(4)}$ and $K_a(z)=2u_a z$, so the <conformal factor> cancels and $\mathcal L_{K_a}g=0$.
Hence the maximally extended <Anti-de Sitter spacetime> has connected <isometry> group locally $SO_0(3,2)$, with the appropriate covering group if one unwraps its time coordinate. The displayed coordinates cover only a patch. The ten local generators do not all give globally complete flows preserving that patch; for example special conformal flows can cross its horizon. The manifest complete patch symmetries include the boundary Poincare transformations and positive dilations. This distinguishes local maximal symmetry from the global <isometries> of a chosen coordinate domain.
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