Klein quadric

Description

Let $F$ be a field. The Klein quadric is the quadric in $\mathbb{P}^5(F)$, with coordinates $[p_{01} : p_{02} : p_{03} : p_{12} : p_{13} : p_{23}]$, given by $$p_{01} p_{23} - p_{02} p_{13} + p_{03} p_{12} = 0.$$ Sending a line of $\mathbb{P}^3(F)$ to its Plücker coordinates is a bijection from the set of lines of $\mathbb{P}^3(F)$ onto the Klein quadric. This bijection is the Klein correspondence.

Two lines of $\mathbb{P}^3(F)$ meet if and only if the line of $\mathbb{P}^5(F)$ joining their images lies entirely on the Klein quadric. Check on an example: the lines $L_1$ through $e_0, e_1$ and $L_2$ through $e_0, e_2$ meet at $[e_0]$. Their images are $[1 : 0 : 0 : 0 : 0 : 0]$ and $[0 : 1 : 0 : 0 : 0 : 0]$. Every point $[s : t : 0 : 0 : 0 : 0]$ of the joining line satisfies the equation, since each product in it has a zero factor.