  In this paper we consider simply connected Lie groups equipped with left
invariant Randers metrics which arise from left invariant Riemannian metrics
and left invariant vector fields. Then we study the intersection between
automorphism and isometry groups of these spaces. Finally it has shown that for
any left invariant vector field, in a special case, the Lie group admits a left
invariant Randers metric such that this intersection is a maximal compact
subgroup of the group of automorphisms with respect to which the considered
vector field is invariant.
