Adjoint orbits of the Lie algebra $sl(2)$

Question

So I have been trying to figure out the ad-joint and co-adjoint orbits of the lie algebra $sl(2)$, I found online that they are supposed to be hyperboloids but I can't seem to get that using my matrix representation. I let my basis for $sl(2)$ to be $$A_1 = \begin{bmatrix} 0 \;\;\; 1\\ 0 \;\;\; 0 \end{bmatrix}, A_2 = \begin{bmatrix} 1 \;\;\;\;\;\; 0\\ 0 \;\;\; -1 \end{bmatrix} A_3 = \begin{bmatrix} 0 \;\;\; 0\\ 1 \;\;\; 0 \end{bmatrix}$$

now if $X \in SL(2)$ then $ X = \begin{bmatrix} a \;\;\; b\\ c \;\;\; d \end{bmatrix}$ and $ad-bc = 1$ and the matrix representation of the ad-joint action is

$$Ad_X = \begin{bmatrix} a^2 \;\;\; -2ab \;\;\; -b^2 \\ -ac \;\;\; ad+bc \;\;\;\; bd \\ -c^2 \;\;\;\; 2dc \;\;\;\; d^2 \end{bmatrix}$$

So I tried to look at the orbit of $A_1$ and using the matrix I get

$$\begin{bmatrix} a^2 \;\;\; -2ab \;\;\; -b^2 \\ -ac \;\;\; ad+bc \;\;\;\; bd \\ -c^2 \;\;\;\; 2dc \;\;\;\; d^2 \end{bmatrix} \cdot \begin{bmatrix} 1 \\ 0 \\ 0 \\ \end{bmatrix} = \begin{bmatrix} a^2\\ -ac \\ -c^2 \end{bmatrix}$$

now I have seen online the orbits are hyperboloids, so I have been trying a mixture of equations but I cannot get my vector to satisfy any of the equations $x^2+y^2-z^2 = R^2$ or any sign change of that equations either. I am very confident in my matrix representation but I just do not see how these orbits satisfy these equations, maybe it's because of my basis choice but I am pretty confident what I have is correct I am just not seeing it, any help is appreciated.

Answer 1

There is only one step between you get the final answer. In the case you calculate you could find that $y^2+xz=0$.This is just a cone, which is also a quadratic curve and you may easily imagine that it's special case of a hyperboloid. Considering the orbit of $\left[\begin{array}{c}\alpha\\\beta\\\gamma\end{array}\right]$, you will find that they satisfy $y^2+xz=\beta^2+\gamma\alpha$ (remember making use of $ad-bc=1$) and is just hyperboloid.

(Well I suggest that you should know when a quadratic curve is a hyperboloid, by converting it into standard quadratic form)