I want to prove that
Consider the map $f: \mathbb{R}^3 \to \mathbb{R}^6$ given by
$$f(x,y,z)=(x^2,y^2,z^2, \sqrt{2} yz, \sqrt{2} xz , \sqrt{2} xy)$$
Then, $f(S^2) \subseteq \mathbb{R^6}$ is a Euclidean submanifold. I want to prove this showing that $f$ is an embbeding. I can prove that $f$ is an immersion, is $f$ an embbeding?
It is not true that $f\mid_{S^2}$ is an embedding: We always have $f(-\xi) = f(\xi)$.
However, as user10354138 says in his comment, $f$ induces an embedding $F : \mathbb RP^2 \to \mathbb R^6$. See Homeomorphism from $P^2\mathbb{R}$ onto the image of $\mathbb{S}^2$ through the Veronese map.
Therefore $f(S^2) = F(\mathbb RP^2)$ is a Euclidean submanifold of $\mathbb R^6$.