Consider the Lie algebra $\mathfrak sl(2,\mathbb R)$ of traceless $2\times 2$ matrices. Suppose $\mathfrak l\subseteq \mathfrak sl (2,\mathbb R)$ is a Lie sub algebra spanned by two elements $\{H,Z\}$ with the property that $[H,Z]=2Z$. I have to show that the Jordan canonical form of $Z$ is given by the matrix $X$ given by $$X = \begin{pmatrix}0 & 1 \\ 0 & 0\end{pmatrix}.$$ My attempt: Observe that the Jordan normal form of a $2\times 2$ matrix $Z$ is one of the following: $$\begin{pmatrix}\lambda_1 & 0 \\ 0 & \lambda_2\end{pmatrix},\ \begin{pmatrix}\lambda_3 & 1 \\ 0 & \lambda_3\end{pmatrix}$$ here $\lambda_1\neq \lambda_2$ and $\lambda_i$ are the Eigen values of $Z$. Since $Z$ is a traceless matrix it follows that the Jordan normal form of $Z$ is given by either $$J_1 = \begin{pmatrix}\lambda & 0 \\ 0 & -\lambda\end{pmatrix},\text{ or }J_2 = \begin{pmatrix}0 & 1 \\ 0 & 0\end{pmatrix}.$$
Now I get stuck, I want to use the fact that $[H,Z]= 2Z$ to conclude that the first option cannot occur, however I don't really know how to do it. I know that $Z = PJ_iP^{-1}$ for $i=1$ or $i=2$. But I don't know how to go from there. Any tips or pushes into the right direction would be much appreciated.
