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
$$-X^2_0-X^2_{d+1}+sum^d_{i=1}dX^2_i=-R^2,quad Rinmathbb{R}$$
with the induced metric. If we compute this induced metric in the coordinates $(t,rho,y^i)$ $(i=1,dots,d)$ defined by
$$begin{cases}
X_0 &= Rcosh{rho}sin{t/R},
X_{d+1} &= Rcosh{rho}cos{t/R},
X_i &= Ry^isinh{rho},
end{cases}$$
we get
$$ds^2 = -cosh^2rho dt^2+R^2drho^2+R^2sinh^2rho dOmega^2_{d-1}$$
with $dOmega^2_{d-1}$ 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 ?