We　give　the　explicit　formulas　of　the　flag　curvature　of　(alpha,　beta)-metrics　of　Berwald　type,　and　correct　an　error　of　the　first　and　third　authors　in　a　previous　article.　Then　we　prove　that　at　any　point　of　a　connected　noncommutative　nilpotent　Lie　group,　the　flag　curvature　of　any　left　invariant　(alpha,　beta)-metric　of　Douglas　type　admits　zero,　positive　and　negative　values,　generalizing　a　theorem　of　Wolf.　Moreover,　we　study　left　invariant　(alpha,　beta)-metrics　of　Douglas　type　on　two　interesting　families　of　Lie　groups　considered　by　Milnor　and　Kaiser,　including　Heisenberg　Lie　groups.　On　these　spaces,　we　present　some　necessary　and　sufficient　conditions　for　(alpha,　beta)-metrics　to　be　of　Berwald　type,　as　well　as　some　necessary　and　sufficient　conditions　for　Randers　metrics　to　be　of　Douglas　type.　We　show　that　every　left　invariant　(alpha,　beta)-metric　of　Douglas　type　on　G　is　an　element　of　G(1),　the　family　which　is　defined　by　Milnor,　is　a　locally　projectively　flat　Randers　metric.　We　also　give　the　explicit　formulas　of　the　flag　curvature　of　left　invariant　Randers　metrics　of　Douglas　type　on　these　spaces　and　show　that,　under　a　condition,　the　flag　curvature　is　negative.