a classical result that, if $n>2$, every restricted conformal transformation of $\mathcal{E}^{1,n-1}$ is induced by a unique element of $\mathrm{A}^{\uparrow}_+(2,n)$ [@DNF; @Fr]. For this reason, we call $\mathrm{A}^{\uparrow}_+(2,n)$ the [*restricted conformal group of the $n$-dimensional Einstein universe*]{}.

[To distinguish the connected component of the identity we proceed as in [@GS] : we consider the cone $\mathcal{C}\subset \bigwedge^2(\R^{n+2})$ of all isotropic bi-vectors, ie the non-zero decomposable elements $V\wedge W$ of $\bigwedge^2(\R^{n+2})$ such that $\langle V,V\rangle = \langle W,W\rangle = \langle V,W\rangle = 0$. The function $$\mathfrak{V}: V\wedge W \in \mathcal{C} \to dV(V,W,E_1,\dots E_n,E_{n+1}-E_0)\in \R$$ never vanishes and the half cones $$\mathcal{C}_+=\{V\wedge W\in \mathcal{C} : \mathfrak{V}(V\wedge W)>0\},\quad  \mathcal{C}_-=\{V\wedge W\in \mathcal{} : \mathfrak{V}(V\wedge W)<0\}$$ are the two connected components of $\mathcal{C}$. Then, $\mathrm{A}^{\uparrow}_+(2,n)$ is the group of all pseudo-orthogonal matrices $\mathbf{B}$ of the scalar product (\[sp\]) such that $\mathrm{det}(\mathbf{B})=1$ and $\mathbf{B}\cdot \mathcal{C}_+=\mathcal{C}_+$.]{}

We put $$\label{mm}\mathtt{m}=(\mathtt{m}_{ji}),\quad \mathtt{m}_{ji}=\langle \mathrm{E}_j,\mathrm{E}_i\rangle,\quad i,j=0,\dots, n+1.$$ Then the column vectors $B_0,\dots B_{n+1}$ of a matrix $\mathbf{B}\in \mathrm{A}^{\uparrow}_+(2,n)$ constitute a [*light-cone basis*]{} of $\R^{n+2}$, ie a positive-oriented basis such that $$\langle B_i,B_j\rangle = \mathtt{m}_{ji}, \quad i,j=0,\dots n+1,\quad B_0\wedge(B_1+B_2)\in \mathcal{C}_+.$$ This allow us to identify $\mathrm{A}^{\uparrow}_+(2,n)$ and the manifold of all light-cone basis of $\R^{n+2}$. We use the notation $\R^{2,n}$ to denote 