It is a well-known fact that $\mathbb C$ has the group structure with respect to the sum. Identifying $\mathbb C$ with the subgroup $$ U_a=\left\{ \begin{pmatrix} 1 & 0 \\ c & 1 \end{pmatrix}: c \in \mathbb C \right\} \subset \operatorname{SL}_2(\mathbb C), $$ we can express the addition "$a+b \in \mathbb C$" via the product law in the group $\operatorname{SL}_2(\mathbb C)$: $$ \begin{pmatrix} 1 & 0 \\ a & 1 \end{pmatrix}\begin{pmatrix} 1 & 0 \\ b & 1 \end{pmatrix}=\begin{pmatrix} 1 & 0 \\ a+b & 1 \end{pmatrix}. $$
On the other hand, $\mathbb C$ has a multiplicative structure itself. Can we express the product "$a \cdot b \in \mathbb C$" using group properties of $\operatorname{SL}_2(\mathbb C)$? I am asking for $f: U_a \times U_a \to U_a$ such that $$ \left(\begin{pmatrix} 1 & 0 \\ a & 1 \end{pmatrix},\begin{pmatrix} 1 & 0 \\ b & 1 \end{pmatrix}\right)\longmapsto\begin{pmatrix} 1 & 0 \\ ab & 1 \end{pmatrix}. $$
