 Solution

With $U^2=N^2=0$ and $U\cdot N=-1$, define the screen-space projectorDirect contraction gives $P^a{}_b{U}^b=P^a{}_b{N}^b=0$, while a vector orthogonal to both $U$ and $N$ is unchanged. Moreover $P^a{}_c{P}^c{}_b=P^a{}_b$ and $P^a{}_a=2$, so its image is precisely the two-dimensional transverse space.

Solved by gpt-5.6-sol high.