homework and exercises - Finding the Basis vectors of a Killing field vector space

I have solved the Killing vector equations for a 2-sphere and got the following answer. $A,B,C$ are three integration constants as expected.

$$\xi_{\theta}=A \sin{\phi}+B\cos{\phi}$$ $$\xi_{\phi}=\cos{\theta} \sin{\theta}(A \cos{\phi}-B\sin\phi)+c \sin^2{\theta}$$

From this, how do I find the basis elements in terms of the tangent vectors $\frac{\partial}{\partial \phi}$ and $\frac{\partial}{\partial \theta}$? One obvious elements if $\frac{\partial}{\partial \phi}$ itself, as the metric is independent of $\phi$. Another method IMO could be to derive it from the angular momentum generators $x\frac{\partial}{\partial y}-y\frac{\partial}{\partial x}$ etc. Is this correct? Would changing the coordinates of these generators give the right answer?

Lastly, How do I do this for any general metric? Is there a standard procedure to find the killing vector fields, as a basis.

Answer

Got it for the 2 sphere.

$\xi=\xi^{\theta}\partial_{\theta}+\xi^{\phi}\partial_{\phi}=-A L_x + B L_y + C L_z$

A, B, C are the same integration constants as in the question and $L_x, L_y, L_z$ are the angular momentum operators written in the spherical polar coordinates.