On the complex projective space $\mathbb{P}^1_\mathbb{C}$ we have an involution $z\mapsto -\frac{1}{\bar{z}}$. Using this as descent datum we should end up with a real form, which is not split (this can be seen for example by the fact that this real form will have no real points, as the involution mentioned above has no fixed points).
I have tried to calculate this real form naively by looking at the map $\sigma: \mathbb{C}[x,y]\to \mathbb{C}[x,y]$ given by $\sigma(\lambda\cdot x) = \bar \lambda \cdot (-y )$ and $\sigma(\lambda\cdot y) = \bar \lambda \cdot (-x )$ , and then looking at the scheme $Proj(\mathbb{C}[x,y]^\sigma)$ i.e. the graded ring of invariants. Is this approach valid?
My result is that $\mathbb{C}[x,y]^\sigma \cong \mathbb{R}[x-y, i(x+y)] $, which to me seems to be isomorphic to the standard real projective space, yielding a contradiction, so I am wondering where the mistake possibly is.