= Group model of three-dimensional anti-de Sitter spacetime
{title2=$\mathrm{AdS}_3\cong SL(2,\mathbb R)$}
Associate $(u,v,x,y)$ with $X=\begin{pmatrix}u+x&y+v\\y-v&u-x\end{pmatrix}$. The ambient quadratic form is $-\det X$; hence the radius-$\ell$ anti-de Sitter quadric is $X=\ell g$ with $g\in SL(2,\mathbb R)$. The action $X\mapsto AXB^{-1}$ gives $SO_0(2,2)\cong(SL(2,\mathbb R)\times SL(2,\mathbb R))/\{(I,I),(-I,-I)\}$. Its induced <bi-invariant pseudo-Riemannian metric> is $g=\ell^2K/8$, giving $\operatorname{Ric}=-2g/\ell^2$ and cosmological constant $-1/\ell^2$. The antipodal quotient is $PSL(2,\mathbb R)$; unwrapping global time gives the universal covering group instead. The connected component is essential in the displayed orthogonal-group isomorphism.
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