I'm reading over the classification of 3-dimensional complex Lie algebras, and have come to the classification of a particular Lie algebra spanned by $\{x,y,z\}$ satisfying the relations $$[x,y] = y, \; [x,z] = y+z, \; [y,z]=0$$
Seeking to find a concrete example of such a Lie algebra (in particular, a subalgebra of $\mathfrak{gl}(n,\mathbb{C})$), I adapted another example to obtain the subalgebra spanned by \begin{align*} x=\begin{pmatrix} 0 & -1 & 0 \\ 0 & 0 & -1 \\ 0 & 0 & -1 \end{pmatrix}, \; \; \; y=\begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \; \; \; z = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & -1 \\ 0 & 0 & 0 \end{pmatrix} \end{align*} which satisfies the above relations. However, I was wondering if a "nicer" example existed, and/or how to derive such an example.
Edit: I managed to find the following subalgebra of $\mathfrak{gl}(n,\mathbb{C})$, derived by considering the matrices of $\text{ad}(x)$, $\text{ad}(y)$ and $\text{ad}(z)$:
\begin{align*} x=\begin{pmatrix} 0 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{pmatrix}, \; \; \; y=\begin{pmatrix} 0 & 0 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \; \; \; z = \begin{pmatrix} 0 & 0 & 0 \\ -1 & 0 & 0 \\ -1 & 0 & 0 \end{pmatrix} \end{align*}