Expression from generators of Special Linear Groups II I wonder whether one can generate this $t$ matrix form the $A_1$ and $A_2$ matrix below. Here
$$
t=\begin{pmatrix} 1& 1& 0\\ 0& 1& 0\\ 0& 0& 1 \end{pmatrix}
$$
from:
$$
A_1=\begin{pmatrix} 0& 0& 1\\ 1& 0& 0\\ 0& 1& 0 \end{pmatrix},
\text{ 
and   }\;\; 
A_2=\begin{pmatrix} 0& 1& 0\\ -1& 0& 0\\ 0& 0& 1 \end{pmatrix}.
$$

Question: How to generate $t$ from $A_1$ and $A_2$? i.e. So what is the exact expression to make $t=\dots A_1 \dots A_2$ as a product of $A_1$ and $A_2$ matrices?

Notice that $(A_1)^3=1$ and $(A_2)^4=1$.
I learned the basics about those generators and SL(N,Z) from the book Coxeter and Moser on "Generators and relations for discrete groups" (published by Springer, 1957).
Linear algebra and special-linear group experts please help. Thank you! :o)

Let me try again, hopefully not telling any needless lies.
The two matrices $A_1$ and $A_2$ are signed permutation matrices, and the group of all signed permutation matrices is the hyperoctohedral group. So the group generated by $A_1$ and $A_2$ is a subgroup of this hyperoctohedral group, which has order $2^3\cdot 3!$. Now, the matrix $t$ has infinite order, and so it is not contained in any finite group of matrices.
