I'm trying to find the orbits when $SL_2$ operates by conjugation on $\mathfrak{sl}_2=Lie(SL_2)=\{A|\operatorname{tr} A=0\}$.
I have tried to write $X\in sl_2$ and corresponding $AXA^{-1}$ for random $A\in SL_2$ in form of $xE_1+yE_2+zE_3$, where $ E_1= \left[ {\begin{array}{cc} 1 & 0 \\ 0 & -1 \\ \end{array} } \right] $,$ E_2= \left[ {\begin{array}{cc} 0 & 1 \\ 0 & 0 \\ \end{array} } \right] $,$ E_3= \left[ {\begin{array}{cc} 0 & 0 \\ 1 & 0 \\ \end{array} } \right] $, and find that for fixed $x_0$, all $ \left[ {\begin{array}{cc} x_0 & y \\ z & -x_0 \\ \end{array} } \right] $ where $yz\leq x_0^2$ are in one orbit, but I can`t see how to deal with the rest.
The orbits of $\operatorname{GL}_2$ on $\mathfrak{sl}_2$ are labelled by the Jordan type of the matrices, which (because we are acting on traceless matrices) fall into two families: $\begin{pmatrix}\lambda & 0 \\ 0 & -\lambda\end{pmatrix}$ for any $\lambda \in \mathbb{C}$, and also $\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$. The orbits of $SL_2$ should be the same.