Euclidean anti-de Sitter space embedding

Let us take $$\mathbb{R^{d+2}}$$ with the cartesian coordinates $$(X_0,\dots,X_{d+1})$$ and the following metric : $$\begin{equation}\label{equ1} ds^2 = -dX^2_0-dX^2_{d+1}+\sum^d_{i=1}dX^2_i. \end{equation}$$ The $$d+1$$-dimensional anti-de Sitter space $$AdS_{d+1}$$ can be seen as the submanifold defined by the relation $$-X^2_0-X^2_{d+1}+\sum^d_{i=1}dX^2_i=-R^2,\quad R\in\mathbb{R}$$ with the induced metric. If we compute this induced metric in the coordinates $$(t,\rho,y^i)$$ $$(i=1,\dots,d)$$ defined by $$\begin{cases} X_0 &= R\cosh{\rho}\sin{t/R},\\ X_{d+1} &= R\cosh{\rho}\cos{t/R},\\ X_i &= Ry^i\sinh{\rho}, \end{cases}$$ we get $$ds^2 = -\cosh^2\rho dt^2+R^2d\rho^2+R^2\sinh^2\rho d\Omega^2_{d-1}$$ with $$d\Omega^2_{d-1}$$ the euclidean metric on the $$d-1$$-sphere and $$\sum^d_{i=0}(y^i)^2=1$$.

I read that Euclidean $$AdS_{d+1}$$ plays and important role in GR but I can't seem to get the metric expression by doing the same kind of reasoning. What metric of $$\mathbb{R}^{d+1}$$ should I start with and wich hyperbolic coordinates should I take ?