list manipulation - Partition a set into subsets of size $k$

Given a set $\{a_1,a_2,\dots,a_{lk}\}$ and a positive integer $l$ , how can I find all the partitions which includes subsets of size $l$ in Mathematica? For instance, given {1,2,3,4} and l=2, the output should be:

$\text{Partition 1: }\{1,2\},\{3,4\}$

$\text{Partition 2: }\{1,3\},\{2,4\}$

$\text{Partition 3: }\{1,4\},\{2,3\}$

In Mathematica notation:

{

 {{1,2},{3,4}},
 {{1,3},{2,4}},
 {{1,4},{2,3}}
}

Edit: Another example for l=3, and {1,2,3,4,5,6}:

{
 {{1, 2, 3}, {4, 5, 6}}, 
 {{1, 2, 4}, {3, 5, 6}}, 
 {{1, 2, 5}, {3, 4, 6}}, 

 {{1, 2, 6}, {3, 4, 5}},
 {{1, 3, 4}, {2, 5, 6}},
 {{1, 3, 5}, {2, 4, 6}}, 
 {{1, 3, 6}, {2, 4, 5}},
 {{1, 4, 5}, {2, 3, 6}},
 {{1, 4, 6}, {2, 3, 5}},
 {{1, 5, 6}, {2, 3, 4}}
}

Answer

Try with

partitions[list_, l_] := Join @@
  Table[
    {x, ##} & @@@ partitions[list ~Complement~ x, l],
    {x, Subsets[list, {l}, Binomial[Length[list] - 1, l - 1]]}
  ]

partitions[list_, l_] /; Length[list] === l := {{list}}

The list must have a length multiple of l

Comments

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