In physics we learn that we can equip $\mathbb{R}^4$ with a bilinear form: $(\cdot,\cdot): \mathbb{R}^4 \times \mathbb{R}^4 \rightarrow \mathbb{R}$, that we call the „Minkowski Metric“. With respect to the standart basis it is given by: $(x,y):= x_0y_0-x_1y_1-x_2y_2-x_3y_3$.
The „matrix representation“ of this bilinear form with respect to the standart basis is:
$$ \eta:=\begin{pmatrix} 1 &0&0&0\\ 0&-1&0&0\\ 0&0&-1&0\\ 0&0&0&-1 \end{pmatrix} $$
Then $(x,y)=x^T \eta y$. My problem now is that I‘ve never seen a proof that this matrix representation doesn’t change if we perform a change of basis to the coordinate system of another „inertial observer“. This would of course follow easily if we knew that such change of basis are carried out by linear transformations from the Lorentz-Group, which by definition are invariant with respect to the „Minkowski Metrik“. So is there more to this or must I just accept that „inertial reference frames“ are just defined to be exactly these bases that just don’t happen to change the matrix representation above?