This is my first question on this site. I hope it is not inappropriate on MO.

Myers and Steenrod proved 1939 that the isometry group of a Riemannian manifold is a lie group. I add a picture where Kobayashi describes the base idea of this proof.

Now I read in the book "Recent trends about lorentzian geometry" the following: (The reference [6] is Myers and Steenrod).

Sadly the author doesn't give more details. Can anyone tell my why this particular proof of M&S does not work for Pseudo-Riemannian manifolds?

EDIT: Maybe this will clear things up a bit.

(1) My questions is NOT about why theorem 1 (on page 278) fails for Lorentzian manifolds. (Theorem 1 is: If $(M,g)$ is a compact Riemannian manifold, then $Iso(M,g)$ is compact.)

(2) I know that the theorems provided by Kobayashi in his book "Transformation groups in differential geometry" can be applied to Lorentzian manifolds as well (e.g. Thm 5.1). So I wonder why this specific proof of M&S only works for Riemannian manifolds.