  We discuss Cartan-Schouten metrics (Riemannian or pseudo-Riemannian metrics
that are parallel with respect to the Cartan-Schouten canonical connection) on
perfect Lie groups. Applications are foreseen in Information Geometry.
Throughout this work, the tangent bundle TG and the cotangent bundle T*G of a
Lie group G, are always endowed with their Lie group structures induced by the
right trivialization. We show that TG and T*G are isomorphic if G possesses a
biinvariant Riemannian or pseudo-Riemannian metric. We also show that, if on a
perfect Lie group, there exists a Cartan-Schouten metric, then it must be
biinvariant. We compute all such metrics on the cotangent bundles of simple Lie
groups. We further show the following. Endowed with their canonical Lie group
structures, the set of unit dual quaternions is isomorphic to TSU(2), the set
of unit dual split quaternions is isomorphic to T*SL(2,R). The group SE(3) of
special rigid displacements of the Euclidean 3-space is isomorphic to T*SO(3).
The group SE(2,1) of special rigid displacements of the Minkowski 3-space is
isomorphic to T*SO(2,1). Some results on SE(3) by N. Miolane and X. Pennec, and
M. Zefran, V. Kumar and C. Croke, are generalized to SE(2,1) and to T*G, for
any simple Lie group G.
