Is the symplectic group over the rationals $\text{Sp}(2n,\mathbb Q)$ dense on the symplectic group $\text{Sp}(2n,\mathbb R)$ over the reals?

Question

The symplectic group is defined as

$$\text{Sp}(2n,F)=\{M\in M_{2n\times 2n}(F) : M^T\Omega M=\Omega\},$$

where

$$\Omega =\left( \begin{matrix}0&I_n\\-I_n&0\end{matrix}\right).$$

Is the symplectic group over the rationals $\text{Sp}(2n,\mathbb Q)$ dense on the symplectic group over the reals $\text{Sp}(2n,\mathbb R)$?

I am aware that the rational orthogonal group is dense on the real orthogonal group. There appears to be at least two ways to show this, but neither seem obvious to extend to the case of symplectic matrices.

1. By the Cartan–Dieudonné theorem it is possible to show that the rational Householder matrices are dense on the reals Density of orthogonal matrices with rational coefficients

2. By the Cayley formula https://mathoverflow.net/questions/90070/existence-of-rational-orthogonal-matrices

One possible method I have been considering is using the fact that the real symplectic group can be written in terms of generators

$$\text{Sp}(2n,\mathbb R)=D(n)\cup N(n) \cup \{\Omega\}$$ where

$${\displaystyle {\begin{aligned}D(n)&=\left\{\left.{\begin{bmatrix}A&0\\0&(A^{T})^{-1}\end{bmatrix}}\,\right|\,A\in \operatorname {GL} (n,\mathbb {R} )\right\}\\[6pt]N(n)&=\left\{\left.{\begin{bmatrix}I_{n}&B\\0&I_{n}\end{bmatrix}}\,\right|\,B\in \operatorname {Sym} (n)\right\}\end{aligned}}}$$.

Is it sufficient to show that there exists groups $D(n,\mathbb Q)$ and $N(n,\mathbb Q)$ which are each dense in $D(n)$ and $N(n)$ respectively? If so, what is the reasoning for this?

Answer 1

For any field $\mathbb{F}$, the symplectic group $Sp(2n,\mathbb{F})$ is generated by symplectic transvections. These are maps of the form $f_{\alpha,u}$, where $\alpha \in \mathbb{F}$, $u \in \mathbb{F}^{2n}$, and $f_{\alpha,u}(v) = v + \alpha B(v,u) u$ for all $v \in \mathbb{F}^{2n}$. Here $B$ is the alternating bilinear form used to define $Sp(2n,\mathbb{F})$.

It seems to me this should be enough to show that $Sp(2n,\mathbb{Q})$ is dense in $Sp(2n,\mathbb{R})$: for $\alpha \in \mathbb{R}$ and $u \in \mathbb{R}^{2n}$, you can find $\alpha' \in \mathbb{Q}$ and $u' \in \mathbb{Q}^{2n}$ such that $f_{\alpha',u'}$ is arbitrarily close to $f_{\alpha,u}$.