I'm looking for a reference of the following result:
Suppose $G$ is a non trivial subgroup of $\mathbb{Z}^n$ of the form $H \cap \mathbb{Z}^n$ where $H$ is a hyperplane of $\mathbb{R}^n$. Let $rank(G)= l$ and $l + m = n$. Then there exist $\tau_1, \dots, \tau_l$, $\pi_1, \dots, \pi_m \in \mathbb{Z}^n$ such that $G$ is generated by $\tau_1, \dots, \tau_l$ and $$M = [\tau_1, \dots, \tau_l, \pi_1, \dots, \pi_m] \in SL(n, \mathbb{Z}).$$
In the article I'm reading (https://iopscience.iop.org/article/10.1070/SM1991v068n01ABEH001371) this is mentioned as a result from the theory of abelian groups and the author refers to Bourbaki algebra book for more details. I have not been able to find the result there so I am not sure if this is the precise statement.
Additionally, I would like to know if there is any explicit relation between $G$ and $M$. For example, if $G$ admits a basis with vectors of norm less than some positive constant $A$ what can we say of the matrix $M$? Is there a constant $c$ depending only on $n$ such that $\| M\| < cA$ ?
Thanks in advance.