Well then in case anyone needs such a transformation, I've constructed the
(non-unique) transformation T

> t:=angle(f,e);
> q:=e++f;
> p:=(c*f-d*e)/q;
> s:=(c*e+d*f)/(q**2);
> transform T;
> T:=identity rotated t xscaled p yscaled q slanted s shifted (a,b);

This yields T=(a,b,c,d,e,f).

Troy