So I just wanted to see if this was the right definition of the Klein-Beltrami model because there have been some issues with a computation.
The Klein Beltrami model is the set $B(0,1)\subseteq \mathbb{R}^n$ where the metric is defined to be
$$g=\frac{1-r^2+x_1^2}{(1-r^2)^2}dx_1^2+...+\frac{1-r^2+x_n^2}{(1-r^2)^2}dx_1^2$$
Where we define $r^2=x_1^2+...+x_n^2.$ Is this definition correct?
No, unfortunately, there are nonzero coefficients on $dx_idx_j$ for $i\ne j$. I believe the coefficient is $2x_ix_j/(1-r^2)^2$.