Steep of the proof of embedding $\mathbb{RP}^{2}$ into $\mathbb{R}^{4}$

Question

Hi i would like some help to prove that the application $F:\mathbb{S}^{2}\to \mathbb{R}^{4}$, where $F(x,y,z)=(x^{2}-y^{2},xy,xz,yz)$ is an immersion. I noted that $F=\bar{F}\circ i$, where $i:\mathbb{S}^{2}\hookrightarrow \mathbb{R}^{3}$ is an embeddeding,in particular, an immersion and $\bar{F}:\mathbb{R}^{3}\to \mathbb{R}^{4}$ with $\bar{F}(x,y,z)=(x^{2}-y^{2},xy,xz,yz)$. Therefore for the prove that $F$ is an immersion simply show that $\bar{F}$ is an immersion, but $\bar{F}$ has two minors with determinants $2x(x^{2}+y^{2})$ and $2y(x^{2}+y^{2})$ , thus $\bar{F}$ has rank $3$ except when $x=y=0$. I don't know continue, any tips?