I am stuck with the following exercise (exercise 15, chapter 9) in O'Neill's book on semi-Riemannian geometry:
"Let $M$ be a complete and connected semi-Riemannian manifold of dimension $n$. Show that the following statements are equivalent:
1) The isometry group of $M$ has dimension $n(n+1)/2$.
2) The algebra of Killing vector fields of $M$ has dimension $n(n+1)/2$.
3) Given any two points $p,q\in M$ and any linear isometry $\Lambda:T_pM\to T_q M$, there exists an isometry $\sigma:M\to M$ such that $\sigma(p)=q$ and $d\sigma_p=\Lambda$."
I have no problem in showing 1) $\Rightarrow$ 2) and 3) $\Rightarrow$ 1), but I cannot show 2) $\Rightarrow$ 3).
More specifically: given 2), it is clear to me that 3) with $p=q$ will hold whenever $\Lambda$ is connected to the identity, but I cannot see why it should hold also for $\Lambda$ not connected to the identity.
Any ideas?