Definition 0.0.1  The scalar product of two four vectors $a,b\in\mathbb{R}^{1,3}$ is simply the four vector dot product.
\[a\cdot b = a_\mu b^\mu = \eta_{\mu\nu}a^\nu b^\mu = \sum_{\mu =0}^3\sum_{\nu=0}^3\eta_{\mu\nu}a^\nu b^\mu = a_0b_0 - \vec{a}\cdot\vec{b}\]