What are the generators of $\mathfrak{gl}_2(\mathbb{C})=\mathfrak{sl}_2(\mathbb{C})\oplus\mathbb{C}$ and their matrix representations?

Question

$[G_1,G_2]=0,\\ [G_1,G_3]=-[G_2,G_3]=G_4,\\ [G_1,G_4]=G_3, [G_2,G_4]=-G_3,\\ [G_3,G_4]=-2G_1+2G_2$

These form the said Lie algebra but how do I find their matrix representations?

Answer 1

\begin{equation}\begin{aligned} G_1 &= \begin{pmatrix}1&0\\0&0\end{pmatrix}\\ G_2 &= \begin{pmatrix}0&0\\0&1\end{pmatrix}\\ G_3 &= \begin{pmatrix}0&1\\1&0\end{pmatrix}\\ G_4 &= \begin{pmatrix}0&1\\-1&0\end{pmatrix} \end{aligned}\end{equation}

I found this by using the standard basis \begin{equation}\begin{aligned} h_1 &= \begin{pmatrix}1&0\\0&0\end{pmatrix}\\ h_2 &= \begin{pmatrix}0&0\\0&1\end{pmatrix}\\ e &= \begin{pmatrix}0&1\\0&0\end{pmatrix}\\ f &= \begin{pmatrix}0&0\\1&0\end{pmatrix} \end{aligned}\end{equation} Computing commutators, it's not much thought after that