$\begin{aligned} & (\alpha \times(\beta+\gamma)) \times(\beta \times \gamma) \cdot(\beta-\gamma) \\ = & (\alpha \times(\beta+\gamma)) \cdot[(\beta \times \gamma) \times(\beta-\gamma)] \\ = & (\alpha \times(\beta+\gamma)) \cdot[(\beta \cdot(\beta-\gamma) \gamma-(\gamma \cdot(\beta-\gamma) \beta] \\ = & (\alpha \times(\beta+\gamma)) \cdot\left[\left(\left.\beta\right|^2-\beta \cdot \gamma\right) \gamma-\left(\gamma \cdot \beta-|\gamma|^2\right) \beta\right] \\ = & (\alpha \times(\beta+\gamma)) \cdot[(1-\beta \cdot \gamma)](\gamma+\beta] \\ = & (1-\beta \cdot \gamma)[\alpha \times \beta) \cdot \gamma+(\alpha \times \gamma) \cdot \beta] \\ \Rightarrow & (1-\beta \cdot \gamma)[(\alpha \times \beta) \cdot \gamma-(\alpha \times \beta) \cdot \gamma]=0\end{aligned}$