In the book "Lectures on Differential Geometry" by Sternberg page 233
"Given a representation,p, of a Lie group G (in particular the adjoint representation) on a vector space F, if p(x) is compact then there exist an invariant scalar product <.,.> on F satisfying = , for all x e G, v, w e V " ie G has bi-invariant metric in the case of p(x) is adjoint represantation. Why p(x) must be compact??? for adjoint representation, what is intuitive reason for non-existance of bi-invariant metric for SE(n) n>1 Lie group? suppose a rigid body in 3d space. attitude of this rigid body can be measured with respect to the inertial reference frame (right invariant) and body frame (left invariant). if this rigid body rotate by rotation matrix R1 and translate by t1 what happen to left invariant metric and right invariant metric and why these quantities are not equal!?