Question about Hartshorne problem (II 4.7)

I am in a weird situation with problem II 4.7 in Hartshorne's Algebraic Geometry. I can do parts b, c,d and e and I understand what to do in part a. but I am stuck on technical details.

Here is the set up. We have a variety over $\mathbb{C}$ and a degree $2$ automorphism $\sigma : X \to X$ which conjugates coefficients in $\mathbb{C}$. We want to produce a variety $X_0$ over $\mathbb{R}$ which base changes to $X$. Let $G = \mathbb{Z} / 2$. Then we have a homomorphism $G \to {\rm Aut}(X)$ which sends $1$ to $\sigma$. The scheme $X_0$ should be $X / G$. The fact that $X$ is separated and that any two points are contained in an affine subscheme guarantee that $X/G$ exists as a scheme. Also it is obvious that $X/G$ is finite type over $\mathbb{R}$.

Problem 1: How does one go about proving that such a quotient is separated?

Problem 2: How do you prove that $X_0$ base changes to $X$?

If the equations of $X$ all have coefficients in $\mathbb{R}$ and $\sigma$ is just conjugation, then problem 2 is easy. Here is an example of when $\sigma$ is not just complex conjugation

Example: Define $\sigma : \mathbb{A}^2_{\mathbb{C}} \to \mathbb{A}^2_{\mathbb{C}}$ to be the morphism whose pullback is $a X^p Y^p \mapsto \overline{a} Y^p X^q$. In this case $\mathbb{A}^2 / G$ is the spectrum of $\mathbb{R}[X+Y,XY,i(X-Y)]$.

I think it is possible to describe the functor $X\mapsto X_0$ very explicitly on affine schemes. This allows us to easily calculate the base change $(X_0)_\mathbb{C}$, and it should also let us prove separatedness, by looking at the morphism $X_0 \to X_0 \times_\mathbb{R} X_0$, and showing that it has closed image by explicitly writing down the ideal sheaf in terms of the corresponding one for $X \to X \times_\mathbb{C} X$.
On an affine patch, say $X = \operatorname{Spec} A$, where $A=\mathbb{C}[X_1,\ldots,X_n]/I$, and we have an involution $\sigma^* : A\to A$ which conjugates $\mathbb{C}$. Let $B$ denote the subring of elements $a\in A$ with $a=\sigma^* a$. Then our $X_0$ should just be $\operatorname{Spec} B$.
There are certain elements that we know are in $B$, namely $X_i + \sigma^* X_i$, and $i(X_i - \sigma^* X_i)$. I believe that it should be possible to use standard techniques of symmetric functions to show that $B$ is generated over $\mathbb{R}$ by these $2n$ elements. (Note, for example, that $X_i (\sigma^* X_i) = ((X_i+\sigma^* X_i)^2 + (i(X_i-\sigma^* X_i))^2)/4$.)
So we can write $B\cong \mathbb{R}[U_1,V_1,\ldots,U_n,V_n]/J$, where $J$ has some nice description in terms of $I$.