Laplacian on coset spaces Edited after @J. Martel's comment: Let us consider the sphere $S^n$ (embedded in $\mathbb{R}^{n+1}$). We know that if $X_i$ represent the vector fields on $S^2$ giving the rotation about the $x_i$-axis, then the Laplacian on $S^2$ is given by $X^2_1 + X^2_2 + X^2_3$. My question is:
Can we have a similar expression for higher dimensional spheres? Specifically, if $Y_i$ generate the Lie algebra of $SO(n+1)$, can we conclude how the Laplacian would look like on $S^n$ from the fact that one can write $S^n$ as a coset space $SO(n+1)/SO(n)$?
Any help will be greatly appreciated. If there is a more general way of answering this question for coset spaces like $G/K$, I would be glad to know about it. Thanks...
 A: I'll give an answer in a few parts:
First, for the $n$-sphere $S^n\subset\mathbb{R}^{n+1}$:  The obvious thing to do is to consider the vector fields
$$
X_{ij} = x_i\frac{\partial }{\partial x_j} - x_j\frac{\partial }{\partial x_i}\ ,
\qquad 0\le i < j\le n.
$$
Then one easily computes that the operator
$$
L = \sum_{0\le i < j\le n}  {X_{ij}}^2
$$
is equal to the Laplacian for the induced metric on $S^n$.  
Second, notice that one doesn't always have uniqueness of this representation.  For example, when $n=3$, consider the vector fields
$$
Y_1^\pm = X_{01}\pm X_{23},\quad Y_2^\pm = X_{02}\pm X_{31},\quad Y_3^\pm  = X_{03}\pm X_{12}.
$$
Then, for functions on $S^3$, one has that the Laplacian equals
$$
 L = (Y_1^+)^2 + (Y_2^+)^2 + (Y_3^+)^2 = (Y_1^-)^2 + (Y_2^-)^2 + (Y_3^-)^2,
$$
Third, in the general case of a Riemannian homogeneous space $M=G/K$, where $G$ acts effectively on $M$ and fixes a metric $g$ on $M$, I gather that the question is whether there always exists a quadratic polynomial $\lambda\in S^2({\frak{g}})$ (where $\frak{g}$ is the Lie algebra of $G$, thought of as $g$-Killing vector fields on $M$) such that $\lambda$, when regarded as a self-adjoint second-order differential operator on $M$, is equal to the Laplacian of the metric $g$. 
This certainly is true in a large number of cases.  For example, if $G$ is semi-simple and $G/K$ is a Riemannian symmetric space, this is true.  Note, though, that $\lambda$ may not be 'positive definite' in the sense that it may not be the sum of squares of a basis of $\frak{g}$.  For example, see the $S^3$ case above.  As another example, if $G=\mathrm{SO}(2,1)$ and $K = \mathrm{SO}(2)$, then $G/K$ is the Poincaré disk, and $\lambda$ in this case turns out to be of the form ${X_1}^2 + {X_2}^2 - {X_3}^2$ for an appropriate basis $X_i$ of ${\frak{so}}(2,1)$.
Whether it is true in all cases (and how unique the representation is) is not clear to me at first glance, but I'll think about it if I get the chance.  It seems that it probably is true when $M$ is compact, or when $M$ is isotropy irreducible, but I should check to be sure before attempting a definitive answer.
