Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 301 / 4 / ii

A Lorentz transformation preserves the quadratic form of the Minkowski metric for every four-vector. Substituting $x^{\prime \sigma }={\Lambda }^{\sigma }{}_{\mu }{x}^{\mu }$ givesThe coefficient difference is a symmetric matrix, and a symmetric quadratic form that vanishes on every vector is zero: evaluating it on basis vectors and their sums recovers each coefficient. ConsequentlyThis is the defining metric-preservation relation of the Lorentz group.