This form is not really an inner product in the common modern definition, because it is not positive definite, only a symmetric bilinear form.

By default, we will use the time negative representation unless stated otherwise:
but another equivalent one is to use a time positive representation:
The matrix is typically denoted by the Greek letter eta.

$η_{μν}=⎣⎢⎢⎢⎡ −1000 0100 0010 0001 ⎦⎥⎥⎥⎤ $

$η_{μν}=⎣⎢⎢⎢⎡ 1000 0−100 00−10 000−1 ⎦⎥⎥⎥⎤ $