I'm trying to understand the proof of Proposition 2.2.10 in Gille-Samuely's book "Central simple algebras and Galois cohomology", and I believe it has an error.

Here's the setup (which I'll try to write in such a way that you won't need to book to follow this). Let $k$ be an infinite field and let $D$ be a central division algebra over $k$ of degree $n$. We then have $D \otimes_k \overline{k} \cong M_n(\overline{k})$. We can identify $M_n(\overline{k})$ with $\mathbb{A}^{n^2}_{\overline{k}}$. Let $U \subset \mathbb{A}^{n^2}_{\overline{k}}$ be the open set consisting of matrices whose characteristic polynomial is separable.

What the authors now want to argue is that $U$ contains a point of $D$, i.e. a $k$-rational point. Their argument for this seems fishy: they say that we can identify $D$ with $\mathbb{A}^{n^2}_k \subset \mathbb{A}^{n^2}_{\overline{k}}$, which makes no sense! The $k$-structure we are considering is not the obvious one.

How can I fix this?

I'm trying to understand the proof of Proposition 2.2.10 in Gille-Samuely's book "Central simple algebras and Galois cohomology" (2nd edition), and I believe it has an error.

Here's the setup (which I'll try to write in such a way that you won't need the book to follow this). Let $k$ be an infinite field and let $D$ be a central division algebra over $k$ of degree $n$. We then have $D \otimes_k \overline{k} \cong M_n(\overline{k})$. We can identify $M_n(\overline{k})$ with $\mathbb{A}^{n^2}_{\overline{k}}$. Let $U \subset \mathbb{A}^{n^2}_{\overline{k}}$ be the open set consisting of matrices whose characteristic polynomial is separable.

What the authors now want to argue is that $U$ contains a point of $D$, i.e. a $k$-rational point. Their argument for this seems fishy: they say that we can identify $D$ with $\mathbb{A}^{n^2}_k \subset \mathbb{A}^{n^2}_{\overline{k}}$, which makes no sense! The $k$-structure we are considering is not the obvious one.

How can I fix this?