The basis of a subgroup $H \subset \mathbb Z^4$ Dear fellow mathstack exchangers,
There is a question in my Abstract Algebra syllabus I have trouble with solving, the main problem is is that there is no clear description of the algorithm I need. Sorry for my lack 
of initiative, but I don`t know where to start. Here is the question:
Find the basis for the subgroup $H \subset \mathbb Z^4$, given by 
$$H = \{ (a,b,c,d) \mid a + b + c + d = 0~~~\text{and}~~a \equiv c~~ (\bmod 12) \}$$
Thank you for the help!
PS: This is my first mathstack exchange post

The relations $a+b+c+d=0$ and $a=c \mod 12$ on $(a,b,c,d)$ can be written as $a+b+a+12e+d=0$ on $(a,b,a+12e,d)$.
In view of it  $g_1:=(1,0,1,-2), g_2:=(0,1,0,-1),$ and $ g_3:=(0,0,12,-12)$ form a basis  of the module $\mathbb{H}$.There are many other bases of $\mathbb{H}.$ PS. The set of the bases of $\mathbb{H}$ is infinite. I don't see any general algorithm to this end.
