It is known that the vector fields $\partial_t$, $\partial_\phi$, $\sin \phi \partial_\theta+\cot \theta \cos\phi \partial_\phi$, and $\cos\phi \partial_\theta-\cot \theta\sin \phi\partial_\phi$ are Killing vector fields of Schwarzschild. $\partial_t$ is evidently Killing since the metric components of the Schwarzschild metric do not depend and $t$, and the remaining Killing vector fields are the Killing vector fields on the sphere.
A priori, it is not clear why these generate the entire Lie algebra Killing vector fields. How can we know that indeed these vector fields generate this Lie algebra?