  In this paper, we study the left invariant spray geometry on a connected Lie
group. Using the technique of invariant frames, we find the ordinary
differential equations on the Lie algebra describing for a left invariant spray
structure the linearly parallel translations along a geodesic and the
nonlinearly parallel translations along a smooth curve. In these equations, the
connection operator plays an important role. Using linearly parallel
translations, we provide alternative interpretations or proofs for some
homogeneous curvature formulae. Concerning the nonlinearly ones, we propose two
questions in left invariant spray geometry. One question generalizes Landsberg
Problem in Finsler geometry, and the other concerns the restricted holonomy
group.
