Regarding the post:
embedding of $\mathbb{RP}^2$ in $\mathbb{R}^4$
I want to understand why $F$ is an immersion. Since $\mathbb{R}P^{2}$ is the quotient of $\mathbb{S}^{2}$ by identifying the antipodal points and we have it suffices to show that the map $f: \mathbb{S}^{2} \subset \mathbb{R}^{3} \rightarrow \mathbb{R}^{4}$ given by $f(x,y,z)=(x^{2}-y^{2},yz,xz,xy)$ is an immersion right? because we have that $F \circ \pi= f$ where $\pi: \mathbb{S}^{2} \rightarrow \mathbb{R}P^{2} = \mathbb{S}^{2}$/~ is the projection map.
OK so I compute the Jacobian and get:
$\begin{bmatrix} 2x & -2y & 0 \\\ 0 & z &y \\ z & 0 & x \\ y & 0 & 0 \end{bmatrix}$
Would it suffice then to show this matrix has rank $3$? If this is not correct can you please explain why $F$ is an immersion? or any other approach is appreciated.