matrix congruence and smith normal form

Question

Fixed $n \geq 2$ and consider $A,B \in GL(n,\mathbb{Z}).$ We know that we have the Smith normal form. One can find $U, V \in SL(n,\mathbb{Z})$ such that $A=UDV.$ So as $B$. The Smith normal form is easy to compute by using Mathematica.

We also define Two matrices $A,B$ are congruent if there exists $X \in PSL(n,\mathbb{Z})$ such that $X^TAX=B$. One can see the related question here.

Now we have the following, if two matrices are congruent, then they have the same $D$. But now I want to ask is that I have two 4 by 4 matrices. I have checked they have the same $D$ in the smith normal form. But how do I know how to verify they are congruent. If it is, I have to find such $X.$

To be more specific, suppose $A=\left( \begin{array}{cccc} 2 & -1 & -1 & -1 \\ -1 & 2 & 0 & 0 \\ -1 & 0 & 2 & 0 \\ -1 & 0 & 0 & 2 \\ \end{array} \right), B=\left( \begin{array}{cccc} 4 & -6 & 4 & -2 \\ -6 & 12 & -8 & 4 \\ 4 & -8 & 6 & -3 \\ -2 & 4 & -3 & 2 \\ \end{array} \right).$ I want to ask whether $A,B$ are congruent. I listed what I already tried.

1. $A$, $B$ both have the same $D=\left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \\ \end{array} \right)$ in the smith normal form. So they define the same integer lattices. But I do not know whether they are congruent.
2. Both of them have 4 positive eigenvalues. So Sylvester theorem show that they are real congruent. But may not be integer congruent.
3. I try to use Mathematica to find $X$ but it cost a lot.

I suspect that they are congruent but I do have an idea on how to solve them. Any ideas or comments are really appreciated. Btw, $A$ is the Cartanf matrix of $SO(8).$ I think it will not help.

Answer 1

The following matrix $T$ will satisfy our condition: $$ T = \begin{pmatrix} 2 & -2 & 2 & -1 \\ 1 & -2 & 2 & -1 \\ 1 & -1 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{pmatrix}. $$ Here is the Magma computation in case you are interested:

> L := LatticeWithGram(Matrix(Integers(), 4, 4, [2, -1, -1, -1, -1, 2, 0, 0, -1, 
0, 2, 0, -1, 0, 0, 2]));
> LL := LatticeWithGram(Matrix(Integers(), 4, 4, [4, -6, 4, -2, -6, 12, -8, 4, 4, 
-8, 6, -3, -2, 4, -3, 2]));
> IsIsometric(L, LL);
true 
[ 2  1  1  0]
[-2 -2 -1  1]
[ 2  2  1  0]
[-1 -1  0  0]

Because of your convention you have to transpose one more time at the end.