Suppose we have the vector space $\mathbb{R}^4$ and the Lorentz's transformation $f:\mathbb{R}^4\to\mathbb{R}^4$. Consider a inner product $g$ given by: $$g(x,y)=x^1y^1+x^2y^2+x^3y^3-c^2t^1t^2$$ for all $$x=(x^1,x^2,x^3,t^1)$$ and $$y=(y^1,y^2,y^3,t^2)$$ in $\mathbb{R}^4$.
How Minkowski concluded this expression?
Why we need that $$g(f(x),f(y))=g(x,y),\ \forall x,y\in\mathbb{R}^4~?$$
Can we define another invariant scalar product in all inertial system?
Regarding question 1. : Our task is to come up with an invariant scalar product in all inertial systems.
For easiness, I'll set $c=1$ and work in one dimension but the conclusions apply for higher dimensions.
We know that any two inertial frames $S'$ and $S$ are related via lorentz transformation as follows
$\begin{bmatrix}x'\\t'\end{bmatrix}=\gamma \begin{bmatrix}1 & -v\\-v & 1\end{bmatrix} \begin{bmatrix}x\\t\end{bmatrix}$
Where $$\gamma=\dfrac{1}{\sqrt{1-v^2}}$$
In a more compact form, the former matrix equation can be rewritten as
$$\mathbf{\chi'}= \Lambda \chi$$
Usually a scalar product can be written like this $$\mathbf { A^T \eta A}$$
Where $\mathbf{A^T}$ is the transpose of matrix $\mathbf A$, and $\mathbf {\eta}$ is a 2 by 2 matrix.
The invariant product we look for then is $$\mathbf {\chi^T\eta \chi=\chi'^T\eta \chi'}$$
We know all the terms in the equation except $\mathbf \eta$, so our task is to find it so that we are able to find out what the invariant scalar product is.
Given that the coordinate systems are related by $\mathbf{\chi'= \Lambda \chi}$.
If we applied Lorentz transformation for $$\chi'^T\eta \chi'$$ we get $$\mathbf{\chi^T \Lambda^T \eta \Lambda \chi}$$
Therefore for $\mathbf {\chi^T \eta \chi=\chi'^T\eta \chi'}$ to be satisfied it must be the case that $$\mathbf \Lambda^T \eta \Lambda=\eta$$
Solving this equation we get $$\eta = \begin{bmatrix}1 & 0\\0 & -1\end{bmatrix} $$
Therefore the inner product is given by
$$\mathbf {\chi^T\eta \chi}=\begin{bmatrix}x & t\end{bmatrix} \begin{bmatrix}1 & 0\\0 & -1\end{bmatrix} \begin{bmatrix}x\\t\end{bmatrix}=x^2 -t^2$$
Which is the Minkowski metric.
On question 3: Any Four-vector $\mathbf V$ that transforms under lorentz transformation, that is
$$\mathbf{V'=\Lambda V}$$
Has an invariant that is associated with it. Since the spacetime coordinates transform under LT, they're Four-vectors with the spacetime interval as the invariant scalar product associated with them, which is $x^2-t^2=x'^2-t'^2$ in one dimension. There are other four vectors like the Four-momentum or Four-current which have an invariant scalar product associated with them.
