Group model of three-dimensional anti-de Sitter spacetime (${\mathrm{AdS}}_3\cong SL(2,\mathbb{R})$)

Associate $(u,v,x,y)$ with $X=\left(\begin{array}{cc}u+x& y+v\\ y-v& u-x\end{array}\right)$. 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 $X\mapsto AX{B}^{-1}$ gives $S{O}_0(2,2)\cong (SL(2,\mathbb{R})\times SL(2,\mathbb{R}))/\{(I,I),(-I,-I)\}$. Its induced bi-invariant pseudo-Riemannian metric is $g={\ell }^{2}K/8$, giving $\mathrm{Ric}=-2g/{\ell }^2$ 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.