I have been working on the following problem:
How can we get an smooth embedding from $\mathbb{R}P^{2}$ to $\mathbb{R}^{4}$, using the fact that $f : {S}^2 \rightarrow \mathbb{R}^6$ given by $f (x; y; z) = (x^2, y^2, z^2, \sqrt{2}yz, \sqrt{2}xz, \sqrt{2}xy)$ induces an smooth embedding from $\mathbb{R}P^{2}$ to $\mathbb{R}^{6}$.
Observation: I already proved that $f : {S}^2 \rightarrow \mathbb{R}^6$ induces an smooth embedding from $\mathbb{R}P^{2}$ to $\mathbb{R}^{6}$. In fact, this can be found here, if anyone wants to read: Prove that $g$ is a submanifold: $g (t,u,v) = (t^2,u^2,v^2,\sqrt{2}uv, \sqrt{2}tv,\sqrt{2}tu)$ .
I would appreciate if anyone could give me some help. Thanks