Equivalent Killing vector fields via an isometry

Suppose that $(M,g)$ is a complete semi-Riemannian manifold. We say that two Killing vector fields $V$ and $W$ are equivalent if there is $\Phi:M\rightarrow M$ an isometry such that $\Phi_*(V)=W$.

If we call $I(M)$ the isometry group of $M$ and $i(M)$ its Lie algebra (which is identified with Killing vectors fields on $M$), then the equivalent Killing vector fields to a given one $v\in i(M)$ are $Im f_v$, where $f_v:I(M)\rightarrow i(M)$ is given by $f_v(\Phi)=Ad_\Phi(v)$.

I am interesting on knowing when two timelike Killing vector field in the anti De Sitter space $\mathbb{H}^4_1$ are equivalent. I already know that in $\mathbb{H}^2_1$ any two timelike Killing vector fields are equivalent, but a similar result can not be true in $\mathbb{H}^3_1$ nor $\mathbb{H}^4_1$.

• What is the dimension of $\mathbb{H}_1^4$? 4 or 5? Aug 1, 2014 at 0:05

Having said that, perhaps the following is helpful. The answer to your question for AdS${}_4$ is contained in pages 15 and 16. In the bottom half of page 15 you will find a list of the 15 orbits and in the list in page 16 the causal character of the corresponding Killing vectors. The causal character varies in many cases.
For AdS${}_5$, the similar lists are in pages 17 (bottom of the page) through 19.