I had a homework problem that exhibited two Killing vector fields for the 2-sphere, asked me to find a third and then asked me if there are any more. I answered no because the Lie Algebra of the isometry group is 3-dimensional (we are only interested in smooth isometries here).
My question is, how could I have seen this without prior knowledge of $SO(3)$, etc.?
I imagine there are several ways. Here's how I would do it:
The dimension of the isometry group $\mathfrak{I}(M,\mathrm{g})$ of a Riemannian $n$-manifold $(M,\mathrm{g})$ has an upper bound of $\frac{n(n+1)}{2}$. Since $S^2$ with the usual Riemannian metric is complete, the Lie algebra of Killing vectors is isomorphic to the Lie algebra of the isometry group. Thus, we cannot have more than $\frac{2(2+1)}{2}=3$ linearly independent Killing vectors.