If $\gamma$ is a curve in $\mathbb{R}^3$ why is the third equality is true?
$$ \begin{split} \ddot{\alpha}(s(t)) &= \frac{s'(t)^2 \cdot \gamma''(t) - s'(t) \cdot s''(t) \cdot \gamma'(t)}{s'(t)^4} \\ & = \frac{\gamma''(t) \langle \gamma'(t), \gamma'(t) \rangle - \gamma'(t) \langle \gamma''(t), \gamma'(t) \rangle}{\left|\gamma'(t) \right|^4} \\ &= \frac{ \gamma'(t) \times (\gamma''(t) \times \gamma'(t))}{\left| \gamma'(t) \right|^4} \end{split} $$
This has nothing to do with curves. It is just an application of the vector triple product:$$u\times(v\times w)=v\langle u,w\rangle-w\langle u,v\rangle.$$