Can anyone give me a reference which explain the derivation of the partial differential operator expression for the laplacian on the euclidean n-dimensional space and on $S^n$ ?

One generally writes the laplacian on the n-dim euclidean space as a sum of a operator on the radial coordinate and $\frac{1}{r^2}$ times the laplacian on $S^n$. 

And very often the laplacian on $S^n$ is written through a recursion relation.  

I am looking for a reference which shows me the derivations of these. 





***I am adding one more question here in this old thread with respect to the Laplace operator,***





One is familiar from Quantum Theory that each of the angular momentum generators $L_{x,y,z}$ are Killing Fields for the standard metric on $S^2$ and the sum of the squares of these generators gives the Laplacian on R^3.

It seems from some literature that this idea in some sense generalizes.

Vaguely what it seems to me is that for a homogeneous spaces $G/H$ if $K_i$ are the killing fields of $G$ then $\sum_i K_i K_i$ is the Laplacian on $G/H$.  

It would be helpful if one can tell me what is the precise statement that contains the above idea and also what are the caveats and the proof of why it should be so. 

In this context people also talk of the "Casimir Laplacian". What precisely is that?