A general automorphism of $\mathfrak{sl}_2$ is given by $x \mapsto \gamma x \gamma^{-1}$ where $x \in \mathfrak{sl}_2$ is written as a $2\times 2$ matrix and $\gamma \in \mathrm{SL}(2,\mathbb C)$ which is to say that, $$\gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$$ where $a,b,c,d\in \mathbb C$ and $ad-bc = 1.$ I am interested to know the conditions on $a,b,c,d$ such that $\gamma$ is an automorphism of finite order $N$. For it to be finite order this would imply, $$\gamma^N x (\gamma^{-1})^N = x$$
in addition to the existing condition $ad-bc=1$. I'm not sure how to go beyond this to get a concrete form of $\gamma$.