Consider the classic map $F:\mathbb{RP}^2\rightarrow \mathbb{R}^4$ defined by $F[x,y,z]=(x^2-y^2,xy,xz,yz)$. This defines a smooth embedding of $\mathbb{RP}^2$ in $\mathbb{R}^4$. It is clearly a topological embedding.
Now, what is the best way to show such map is an immersion? We can compute $DF$ and note that the matrix will have rank 2, but is there an intuitive geometric way of showing that this topological embedding is actually an immersion?