Let's consider in $\mathbb{R}^4$ the set \begin{align} S=\{(x,y,z,t) \in \mathbb{R}^4: &x^2y^2+y^2z^2+x^2z^2=xyz,\\ &x^2y^2+2x^2z^2+3z^2y^2=xyzt\}. \end{align} I have to prove that $S$ is a submanifold of $\mathbb{R}^4$.
I think I need to show that the inclusion $i: S \rightarrow \mathbb{R}^4$ is an embedding, so $i_{*|p}$ is injective and a homeomorphism with its image. In order to show that $i_{*|p}$ is injective I have to check that the Jacobian of the function $F: \mathbb{R}^4 \rightarrow \mathbb{R}^2, (x,y,z,t) \mapsto (x^2y^2+y^2z^2+x^2z^2-xyz, x^2y^2+2x^2z^2+3z^2y^2-xyzt) $ has rank 2. Am I right?
Is there another way to solve it?