I am a little confused.
In an exercise, I am asked to find all Killing vectors of the product of the Anti de Sitter and two-sphere metric, given by: $$\text{d}s^2=\frac{-\text{d}t^2+\text{d}y^2}{y^2}+\text{d}\theta^2+\sin^2(\theta)\text{d}\phi^2.$$
First of all, I know that the two-sphere as well as the Anti de Sitter space are maximally symmetric. Is it then true that their product is still maximally symmetric? Can we then deduce that there should be exactly 4 independent Killing vectors?
Secondly, I know that the metric is independent of $t$ and $\phi$, so two Killing vectors are easily found. But is there an easy way to find the others?