For a lattice $\Gamma=A\mathbb{Z}^n$ where $A$ is an $n$-dimensional invertible matrix, we define its squared length spectrum to be $L_\Gamma:=\{(\lambda^2,m):0\neq m=\{\gamma\in \Gamma: \lambda^2=\|\gamma\|^2\}\}$.
Let $A=\begin{bmatrix}A_0 & 0\\ 0 & A_1 \end{bmatrix}$ be the matrix of a lattice $\Gamma$. I want to show that if $A_0$ is of dimension $3$ or lower, then any other lattice $\Lambda=A'\mathbb{Z}^n$ that has equal squared length spectrum to that of $\Gamma$ has that $A'$ is a block matrix with the same dimensions as the blocks of $A$ (up to congruency, i.e. an element of the matrix group $O_n(\mathbb{R})$).
The connection to quadratic forms is clear; it is easy to see that $A^TA$ is a positive definite quadratic form that is also a block matrix, with dimensions equal to those of $A$. Meaning that we can look at quadratic forms and their representation numbers to try to solve this problem.
I want to know if such a result exists or if it unknown. It is not true if the dimension of both block are greater than $3$, and there are plenty of examples of that. The motivation for this problem comes from the eigenvalue problem on flat tori.