  We give the explicit formulas of the flag curvatures of left invariant
Matsumoto and Kropina metrics of Berwald type. We can see these formulas are
different from previous results given recently. Using these formulas, we prove
that at any point of an arbitrary connected non-commutative nilpotent Lie
group, the flag curvature of any left invariant Matsumoto and Kropina metrics
of Berwald type admits zero, positive and negative values, this is a
generalization of Wolf's theorem. Then we study $(\alpha,\beta)$-metrics of
Berwald type and also Randers metrics of Douglas type on two interesting
families of Lie groups considered by Milnor and Kaiser, containing Heisenberg
Lie groups. On these spaces, we present some necessary and sufficient
conditions for $(\alpha,\beta)$-metrics to be of Berwald type and also some
necessary and sufficient conditions for Randers metrics to be of Douglas type.
All left invariant non-Berwaldian Randers metrics of Douglas type are given and
the flag curvatures are computed.
