Why is the Fano variety of lines on a smooth three-dimensional quadric isomorphic to $\mathbb{P}^3$?

Question

Let $Q \subset \mathbb{P}^4$ be a smooth three-dimensional quadric over an algebraically closed field $k$ ($\mathrm{char}(k) \neq 2$) and let $F$ be the Fano variety of lines on $Q$.

In "Iskovskikh V. A., Prokhorov Yu. G. - Fano varieties" at page 29 it is stated that $F$ is isomorphic to $\mathbb{P}^3$.

Why is this true?

Answer 1

A proof of this well-known fact is outlined in Harris' book Algebraic Geometry, Exercise 22.6 page 290.

More generally, Lecture 22 in the aforementioned book contains a nice treatment of linear spaces on quadrics hypersurfaces.