# Isometries of Minkowski space [duplicate]

Let $\eta$ be a bi-linear form on $\mathbb{R}^4$ given by $$\eta\left(x,\,y\right)=x_0y_0-x_1y_1-x_2y_2-x_3y_3$$ The Poincare group is defined as the group of isometries of $\eta$, that is, the group of all maps $\Lambda:\mathbb{R}^4\to\mathbb{R}^4$ such that for all $(x,\,y)\in\left(\mathbb{R}^4\right)^2$, $$\eta\left(\Lambda\left(x-y\right),\,\Lambda\left(x-y\right)\right)=\eta\left(x-y,\,x-y\right)$$

Then one usually says that these comprise affine transformations, for example, in the Wikipedia article on the Poincare group: "The Poincaré group itself is the minimal subgroup of the affine group".

My question is: is there a mathematical theorem that says, just given the data I specified so far (or more data?), that $\Lambda$ has to be affine (and if so, what is that theorem and its proof?), or is it part of the physical input which stipulates that lines must be sent to lines?