$\begin{aligned} & A=\left[\begin{array}{cc}1 & -\alpha \\ \alpha & \beta\end{array}\right] \quad A A^{\top}-I_2 \\ & \Rightarrow \quad\left[\begin{array}{cc}1 & -\alpha \\ \alpha & \beta\end{array}\right]\left[\begin{array}{cc}1 & \alpha \\ -\alpha & \beta\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right] \\ & \Rightarrow \quad\left[\begin{array}{cc}1+\alpha^2 & \alpha-\alpha \beta \\ \alpha-\alpha \beta & \alpha^2+\beta^2\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right] \\ & \Rightarrow \quad a^2=0 \& b^2=1 \\ & \therefore \quad a^4+b^4=1\end{aligned}$