LLL Hermite normal form algorithm: 4x4 Example

alpha=1
   P                   A
 1 0 0 0        -6   9 -15 -18 
 0 1 0 0         4  -6  10  12   (1)
 0 0 1 0        10 -15  18  35 
 0 0 0 1       -24  36 -46 -82 
Row 1 -> - Row 1
-1 0 0 0         6  -9  15  18 
 0 1 0 0         4  -6  10  12   (2)
 0 0 1 0        10 -15  18  35 
 0 0 0 1       -24  36 -46 -82 
Swapping Rows 1 and 2
 0 1 0 0         4  -6  10  12      
-1 0 0 0         6  -9  15  18   (3)
 0 0 1 0        10 -15  18  35 
 0 0 0 1       -24  36 -46 -82 
Row 2 -> Row 2 - Row 1
 0  1 0 0        4  -6  10  12 
-1 -1 0 0        2  -3   5   6   (4)     
 0  0 1 0       10 -15  18  35    
 0  0 0 1      -24  36 -46 -82 
Swapping Rows 1 and 2
-1 -1 0 0        2  -3   5   6 
 0  1 0 0        4  -6  10  12   (5)
 0  0 1 0       10 -15  18  35 
 0  0 0 1      -24  36 -46 -82 
Row 2 -> Row 2 - 2 x Row 1
-1 -1 0 0        2  -3   5   6 
 2  3 0 0        0   0   0   0   (6)
 0  0 1 0       10 -15  18  35 
 0  0 0 1      -24  36 -46 -82 
Swapping Rows 1 and 2
 2  3 0 0        0   0   0   0 
-1 -1 0 0        2  -3   5   6   (7)
 0  0 1 0       10 -15  18  35 
 0  0 0 1      -24  36 -46 -82 
Row 3 -> Row 3 - 5 x Row 2
 2  3 0 0        0  0   0   0 
-1 -1 0 0        2 -3   5   6    (8)
 5  5 1 0        0  0  -7   5 
 0  0 0 1      -24 36 -46 -82 
Swapping Rows 2 and 3
 2  3 0 0        0  0   0   0
 5  5 1 0        0  0  -7   5    (9)
-1 -1 0 0        2 -3   5   6
 0  0 0 1      -24 36 -46 -82
Row 2 -> Row 2 + -2 x Row 1
 2  3 0 0        0  0   0   0
 1 -1 1 0        0  0  -7   5    (10)
-1 -1 0 0        2 -3   5   6
 0  0 0 1      -24 36 -46 -82
Row 2 -> - Row 2
 2  3  0 0       0  0   0   0
-1  1 -1 0       0  0   7  -5    (11)
-1 -1  0 0       2 -3   5   6
 0  0  0 1     -24 36 -46 -82
Row 4 -> Row 4 + 12 x Row 3
  2   3  0 0     0  0  0   0
 -1   1 -1 0     0  0  7  -5     (12)
 -1  -1  0 0     2 -3  5   6
-12 -12  0 1     0  0 14 -10
Swapping Rows 3 and 4
  2   3  0 0     0  0  0   0
 -1   1 -1 0     0  0  7  -5     (13)
-12 -12  0 1     0  0 14 -10
 -1  -1  0 0     2 -3  5   6
Row 3 -> Row 3 + -2 x Row 2
  2   3  0 0     0  0 0  0
 -1   1 -1 0     0  0 7 -5       (14)
-10 -14  2 1     0  0 0  0
 -1  -1  0 0     2 -3 5  6
Swapping Rows 2 and 3
  2   3  0 0     0  0 0  0
-10 -14  2 1     0  0 0  0       (15)
 -1   1 -1 0     0  0 7 -5
 -1  -1  0 0     2 -3 5  6
Row 2 -> Row 2 + 5 x Row 1
 2  3  0 0       0  0 0  0
 0  1  2 1       0  0 0  0       (16)
-1  1 -1 0       0  0 7 -5
-1 -1  0 0       2 -3 5  6
Swapping Rows 1 and 2
 0  1  2 1       0  0 0  0
 2  3  0 0       0  0 0  0       (17)
-1  1 -1 0       0  0 7 -5
-1 -1  0 0       2 -3 5  6
The transformation matrix is
-1 -1  0 0
-1  1 -1 0
 0  1  2 1
 2  3  0 0
The row echelon Form is
2 -3 5  6
0  0 7 -5
0  0 0  0
0  0 0  0