Let $S = \big(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \big)$ be a matrix with determinant $ad - bc =1$ (this is the case I'm interested in, but I don't think it is essential). Then the matrix $$ T = \left( \begin{matrix} a^2 & 2ac & c^2 \\ ab & ad+bc & cd \\ b^2 & 2bd & d^2 \end{matrix} \right) $$ occurs in the theory of binary quadratic forms and has determinant $\det T = (\det S)^3$.
My question: Can $T$ be written as a product of three matrices whose determinant is obviously $ad-bc$, such as, for example, the following? $$ \left( \begin{matrix} a & b & 0 \\ c & d & 0 \\ 0 & 0 & 1 \end{matrix} \right) $$ For what it's worth, the map $S \mapsto T$ induces a homomorphism SL$_2({\mathbb Z}) \longrightarrow $ SL$_3({\mathbb Z})$. Preferably, this should be a consequence of a factorization of $T$ as the one I'd like to have. Relevant references are also welcome.
We can do the first step and write $$ T=\begin{pmatrix} a & b & 0 \cr c & d & 0 \cr 0 & 0 & 1\end{pmatrix}\cdot \begin{pmatrix} \frac{a(ad-b^2)}{ad-bc} & \frac{2acd-abd-b^2c}{ad-bc} & \frac{cd(c-b)}{ad-bc} \cr \frac{a^2(b-c)}{ad-bc} & \frac{a(ad+bc-2c^2)}{ad-bc} & \frac{c(ad-c^2)}{ad-bc} \cr b^2 & 2bd & d^2 \end{pmatrix} $$ The matrix $T$ also "arises" in the description of Lie algebra automorphisms of $\mathfrak{sl}_2(K)$ with $S\in GL_2(K)$. But I don't see how this helps.