$P$-separation of variables in Laplace's equation

The $P$-separation of variables in Laplace's equation $\Delta_2u=0$ in flat $n$-dimensional space $S_n$ is proved to be equivalent to the complete separation of variables in the invariant Laplace equation
$$ \Delta u\equiv \left\{\Delta_2+\frac{n-2}{4(n-1)}R\right\}u=0, $$
in a space $V_n$ of constant curvature $K\ne0$ ($\Delta$ is the invariant Laplacian, and $R$ is the scalar curvature, all in $V_n$).