We study the relationship between the signature of a semisimple Lie group and a pseudoRiemannian manifold on wich the group acts topologically transitively and isometrically. We also provide a descrip­tion of the bi-invariant pseudo-Riemannian metrics on a semisimple Lie Group over R in terms of the complexification of the Lie algebra associated to the group, and then we utilize it to prove a remark of Gromov.