Generators of $\text{GL}_2(\mathbb{Z})$ and $\text{SL}_2(\mathbb{Z})$. We denote by $(a,b,c)$ the integral binary quadratic form $q(x,y)=ax^2+bxy+cy^2$. Also, we denote by $\sim$ (resp. $\sim_+$) $\text{GL}_2(\mathbb{Z})$ equivalence (resp. $\text{SL}_2(\mathbb{Z})$ equivalence).
I know that for all $a,b,c\in\mathbb{Z}$ we have that $(a,b,c)\sim (a,-b,c)$ and
$$(a,b,c)\sim_+ (c,-b,a)\sim_+ (a,b+2a,c+b+a)\sim_+ (a,b-2a,c-b+a).$$
This is simply the fact that
$\begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix}$
is in $\text{GL}_2(\mathbb{Z})$ and
$$\begin{bmatrix}0 & 1 \\ -1 & 0\end{bmatrix},\qquad \begin{bmatrix}1 & 1 \\ 0 & 1\end{bmatrix},\qquad\begin{bmatrix}1 & -1 \\ 0 & 1\end{bmatrix}$$
are in $\text{SL}_2(\mathbb{Z})$.
I want to know if all equivalent forms can be obtained by repeated applications of those rules. In other words, whether these matrices generate the respective groups or not.
(Since $\mathbb{Z}$ is an euclidean domain, I know that $\text{GL}_2(\mathbb{Z})$ is generated by elementary matrices but it's not clear to me that all elementary matrices can be written as a product of the ones above.)

It is true that $\mathrm{SL}_2(\mathbb Z)$ is generated as a monoid by
$$S = \begin{bmatrix}0 & 1 \\ -1 & 0\end{bmatrix}, \qquad  T = \begin{bmatrix}1 & 1 \\ 0 & 1\end{bmatrix},\qquad T^{-1} = \begin{bmatrix}1 & -1 \\ 0 & 1\end{bmatrix}$$
and thus $\mathrm{GL}_2(\mathbb Z)$ is generated by those matrices together with $\begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix}$. See for example Modular group at Wikipedia, section "Presentation". I think this is originally due to Poincaré. Serre's book "Trees" contains more of this.
For references, Generators of $\text{GL}_{2}(\mathbb{Z})$ group, good reference book?
