Some matrix procedures also have an option which gives you the 
transformation matrix.

sage: M=MatrixSpace(ZZ,4,4)
sage: A=M.random_element()
sage: A.echelon_form(transformation=True)
(
[   1    0    1 3032]  [  85   31  484    7]
[   0    1    1 2176]  [  61   22  347    5]
[   0    0    2 5636]  [ 158   57  899   13]
[   0    0    0 9095], [ 255   92 1452   21]
)

So an efficient way would be just to multiply with that matrix.

Currently there is sadly enough no such option for LLL reduction so you have 
to find an explicit base change yourself.

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