  We use the technique of invariant frames to study a left invariant spray
structure on a Lie group, and calculate its S-curvature and Riemann curvature,
which generalizes the corresponding formulae in homogeneous Finsler geometry.
Using the canonical bi-invariant Berwald spray structure as the reference, any
left invariant spray structure can be associated with a spray vector field on
the Lie algebra. We find the correspondence between geodesics for a left
invariant spray structure and the inverse integral curves of its spray vector
field. As an application for this correspondence, we provide an alternative
proof for Landsberg Conjecture in the case of homogeneous Finsler surfaces.
