  Lie groups of automorphisms of cotangent bundles of Lie groups are completely
characterized and interesting results are obtained. We give prominence to the
fact that the Lie groups of automorphisms of cotangent bundles of Lie groups
are super symmetric Lie groups. In the cases of orthogonal Lie lgebras,
semi-simple Lie algebras and compact Lie algebras we recover by simple methods
interesting co-homological known results.
  The Lie algebra of prederivations encompasses the one of derivations as a
subalgebra. We find out that Lie algebras of cotangent Lie groups (which are
not semi-simple) of semi-simple Lie groups have the property that all their
prederivations are derivations. This result is an extension of a well known
result due to M\"uller. The structure of the Lie algebra of prederivations of
Lie algebras of cotangent bundles of Lie groups is explored and we have shown
that the Lie algebra of prederivations of Lie algebras of cotangent bundle of
Lie groups are reductive Lie algebras.
  We have studied bi-invariant metrics on cotangent bundles of Lie groups and
their isometries. The Lie algebra of the Lie group of isometries of a
bi-invariant metric on a Lie group is composed with prederivations of the Lie
algebra which are skew-symmetric with respect to the induced orthogonal
structure on the Lie algebra. We have shown that the Lie group of isometries of
any bi-invariant metric on the cotangent bundle of any semi-simple Lie groups
is generated by the exponentials of inner derivations of the Lie cotangent
algebra.
  Last, we have dealt with an introduction to the geometry the Lie group of
affine motions of the real line $\mathbb R$, which is a K\"ahlerian Lie group.
