Having a struggle finding a lie algebra homomorphism between these two spaces. So i know that $\mathbb{C} \cong \lambda\left( \begin{array}{ccc} 1 & 0 \\ 0 & 1 \\ \end{array} \right) $, where $\lambda \in \mathbb{C}$. A basis for $sl(2,\mathbb{C})$ is $ \{\left( \begin{array}{ccc} 1 & 0 \\ 0 & -1 \\ \end{array} \right), \left( \begin{array}{ccc} 0 & 1 \\ 0 & 0 \\ \end{array} \right), \left( \begin{array}{ccc} 0 & 0 \\ 1 & 0 \\ \end{array} \right) \}$.
The dimension of $< \left( \begin{array}{ccc} 1 & 0 \\ 0 & 1 \\ \end{array} \right), \left( \begin{array}{ccc} 1 & 0 \\ 0 & -1 \\ \end{array} \right), \left( \begin{array}{ccc} 0 & 1 \\ 0 & 0 \\ \end{array} \right), \left( \begin{array}{ccc} 0 & 0 \\ 1 & 0 \\ \end{array} \right)>$ is 4, and so this space generates $gl(2,\mathbb{C})$. So these two spaces are isomorphic as vector spaces, but I can't seem to find a function that would induce a lie algebra isomorphic. I tried defining a homomorphism by
$\varphi : gl(2,\mathbb{C}) \rightarrow sl(2,\mathbb{C}) \oplus \mathbb{C}$
by,
$\varphi(A) = (\varphi_1(A),\varphi_2(A))$
where $\varphi_1(A) = \left( \begin{array}{ccc} a & b \\ c & -a \\ \end{array} \right)$
$\varphi_2(A)= tr(A)$
where $ A = \left( \begin{array}{ccc} a & b \\ c & d \\ \end{array} \right) $ but this did not preserve the lie bracket. Any insight as to what this homomorphism would be?