Let $A_1, \cdots, A_n$ be non-singular symmetric matrices with integer entries which are simultaneously diagonalizable. Define the bilinear map
$$\displaystyle [\cdot, \cdot] : \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}^n$$
by
$$[\mathbf{x}, \mathbf{y}] = \begin{bmatrix} \mathbf{x}^T A_1 \mathbf{y} \\ \vdots \\ \mathbf{x}^T A_n \mathbf{y} \end{bmatrix}.$$
By construction, $[\cdot, \cdot]$ is symmetric.
Let $\mathbf{x}^{(1)}, \cdots, \mathbf{x}^{(n)}$ be vectors in $\mathbb{R}^n$ and let $f_j$ be the $(j-1)$-shift operator on $\mathbb{R}^n$. That is,
$$\displaystyle f_j((x_1, \cdots, x_n)^T) = (x_{n-j+2}, \cdots, x_n, x_1, \cdots, x_{n-j+1})^T \text{ for } j = 1, \cdots, n.$$
Now define
$$\mathbf{y}^{(1)} = \mathbf{x}^{(1)} \text{ and } \mathbf{y}^{(j)} = [\mathbf{x}^{(j)}, \mathbf{y}^{(j-1)}]$$
for $j = 2, \cdots, n$. Similarly, define
$$\displaystyle \mathbf{z}^{(1)} = \mathbf{x}^{(1)} \text{ and } \mathbf{z}^{(j)} = [f_j(\mathbf{x}^{(j)}), \mathbf{z}^{(j-1)}]$$
for $j = 2, \cdots, n$.
Consider the variety $V \subset \mathbb{R}^{n^2}$ defined by
$$\displaystyle V = \{(\mathbf{x}^{(1)}, \cdots, \mathbf{x}^{(n)}) \in \mathbb{R}^n \times \cdots \mathbb{R}^n \cong \mathbb{R}^{n^2} : \mathbf{z}^{(n)}, \mathbf{y}^{(n)} \in \Pi\}$$
where $\Pi$ is the plane in $\mathbb{R}^n$ defined by
$$\displaystyle \Pi = \{(x_1, x_2, 0, \cdots, 0) \in \mathbb{R}^n : x_1, x_2 \in \mathbb{R}\}.$$
My question is: under what hypotheses on the matrices $A_1, \cdots, A_n$ can one prove that $V$ has the maximum possible codimension which is $2n - 4$?
Edit: the motivation is as follows. Suppose that $K$ is a cyclic, Galois number field of degree $n$ over $\mathbb{Q}$ and $\mathcal{B} = \{w_1, \cdots, w_n\}$ be a basis of an order $\mathcal{O}$ in $K$ consisting of Galois conjugates. Indeed, suppose that $\sigma$ is a generator of the Galois group of $K$, then the $w_j$'s satisfy $w_j = \sigma^{j-1}(w_1)$ for $j = 1, \cdots, n$. Then the bilinear mapping described above is simply given by multiplication in $K$, with respect to the basis $\mathcal{B}$ and the variety we are looking for is given by a rank-2 submodule of $\mathcal{O}$.
