The most important feature of the inner product and norm squared is that "these are quantities unaffected by Lorentz transformations". In fact, it can be taken as the defining property of a Lorentz transformation that it preserves the inner product (i.e. the value of the corresponding bilinear form on two vectors). This approach is taken more generally for "all" classical groups definable this way in classical group. There, the matrix is identical in the case (the Lorentz group) to the matrix to be displayed below.