  Horizontal endomorphisms, almost complex structures, vertical, horizontal and
complete lifts on prolongation of a Lie algebroid are considered. Then using
exact sequences, semisprays are constructed. Moreover, important geometrical
objects such as classical distinguished connections, torsions and partial
curvatures are studied on prolongation of Lie algebroids. Considering pullback
bundle, covariant derivatives are scrutinized based on anchor map. Several
Finsler geometry models on Lie algebroid structures, will organized via recent
arguments. Finally, it will be overhaulled some special Finsler Lie algebroid
spaces.
