Grouping lists by their first elements and deleting some groups

Question

I have a list

{{1, 4, 1, -3}, {1, 4, 1, -1}, {1, 4, 2, -2}, {1, 5, 2, -4}, {1, 5, 
  2, -1}, {1, 5, 4, -3}, {1, 5, 4, -2}, {1, 6, 3, -5}, {1, 6, 
  3, -1}, {1, 7, 4, -6}, {1, 7, 4, -1}, {1, 8, 5, -7}, {1, 8, 
  5, -1}, {2, 5, 1, -1}, {2, 7, 1, -3}, {2, 7, 3, -1}, {2, 7, 
  4, -2}, {2, 9, 2, -4}, {2, 9, 5, -1}, {3, 6, 1, -1}, {3, 7, 
  2, -1}, {3, 8, 2, -2}, {3, 8, 3, -1}, {3, 9, 4, -2}, {3, 9, 
  4, -1}, {3, 10, 1, -3}, {3, 10, 5, -1}, {4, 7, 1, -1}, {4, 9, 
  3, -1}, {5, 8, 1, -1}, {5, 9, 2, -1}, {5, 10, 3, -1}, {6, 9, 
  1, -1}, {7, 10, 1, -1}}

In the list there are some sets have first three elements equal. For exmaple, {1, 4, 1, -3} and {1, 4, 1, -1}. With two this sets, I want to have {1, 4, 1, -3, -1}. How can I get like that for all sets in list? After that, remove the sets have four elements.

Answer 1

Where list is your list,

Join[#[[1, ;; 3]], Flatten@#[[All, 4 ;;]]] & /@ 
 DeleteCases[GatherBy[list, #[[;; 3]] &], {_List}]

(* {{1, 4, 1, -3, -1}, {1, 5, 2, -4, -1}, {1, 5, 4, -3, -2}, {1, 6, 
  3, -5, -1}, {1, 7, 4, -6, -1}, {1, 8, 5, -7, -1}, {3, 9, 4, -2, -1}} *)

Modifying @ciao 's code:

With[{gb = DeleteCases[GatherBy[#, #[[;; 3]] &], {_List}]}, 
   Flatten /@ 
    Transpose[{gb[[All, 1, ;; 3]], 
      Flatten /@ gb[[All, All, 4 ;;]]}]] &@list

(* {{1, 4, 1, -3, -1}, {1, 5, 2, -4, -1}, {1, 5, 4, -3, -2}, {1, 6, 
  3, -5, -1}, {1, 7, 4, -6, -1}, {1, 8, 5, -7, -1}, {3, 9, 4, -2, -1}} *)

Answer 2

KeyValueMap[
   If[Length[#2] == 1, Nothing, Join[##]] &
] @  GroupBy[data, Most -> Last]

 {{1, 4, 1, -3, -1}, {1, 5, 2, -4, -1}, {1, 5, 4, -3, -2}, {1, 6, 
   3, -5, -1}, {1, 7, 4, -6, -1}, {1, 8, 5, -7, -1}, {3, 9, 4, -2, -1}}