The four acceleration is defined as $$\alpha^\mu = \gamma_V ^4 \left(\frac{\vec{v} \cdot \vec{a}}{c},\frac{\vec{v} \cdot \vec{a}}{c^2} \vec{v} + \frac{1}{\gamma_V ^2} \vec{a} \right)$$ where $\vec{v}$ is ordinary velocity and $\vec{a}$ ordinary acceleration in a certain reference frame.
Our professor told us that it comes straight forward from the definition that $$\alpha_\mu \alpha^\mu = \eta_{\mu \nu} \alpha^\mu \alpha^\nu = -\alpha^0 \alpha^0 + \vec{\alpha} \cdot \vec{\alpha} = |\vec{a}|^2$$ but I can't understand the last passage.