I have a question on $3$-dimensional Lie algebra $L$ over ${\bf C}$ (cf. Erdmann and Wildon's book)
Assume that $$ L=(x,y,z),\ L'=(y,z)$$
Then the book states that there exits two kinds of $L$ :
(1) $$ [y,z]=0,\ {\rm ad}_x = \left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & \mu \\ \end{array} \right) $$ wrt $\{ x,y,z\}$. Here is an example $$ y= \left( \begin{array}{ccc} 0 & 0 & 1\\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} \right),\ z=\left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & \mu \\ 0 & 0 & 0 \\ \end{array} \right),\ x= \left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \\ \end{array} \right) $$
(2) $$ [y,z]=0,\ {\rm ad}_x = \left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \\ \end{array} \right) $$ wrt $\{ x,y,z\}$.
${\bf Question :}$ Give me an example of this kind.
Thank you.