TITLE: Embedding $RP^2$ into $R^4$
QUESTION [7 upvotes]: I have a homework Question which asks to show that the map
\begin{equation}f:R^3\rightarrow R^4, f(x,y,z)=(x^2-y^2,xy,xz,yz)\end{equation} 
Induces an embedding of $RP^2$ into $R^4$
Overall I have a fairly good idea of how I want to go about showing this, however I am looking for a "neat" way of showing $f$ is injective (take neat to mean whatever you want in this setting). 
I have an idea that is probably far reaching, but I was wondering if we could somehow use the kernel of $f$ in our argument? If we had a group homomorphism that would be one thing, but this is not so. However it does seem like the only element that gets mapped to zero is $(0,0,1)$ subject to the domain $S^2$, so maybe that's something? Honestly I'm just really lazy and don't want to brute force algebra onto this function to see that it is injective. 
I am just looking for some good hints here, anything is appreciated!

REPLY [2 votes]: See Section 2 Example of (Embedding of $RP^2$ into $R^4$): 
https://www.cs.uic.edu/~sxie/lecture_notes/diff_manifolds/lecture10.pdf