Let $\sigma =\begin{pmatrix} 0 & 1 \\ -1 & 0\end{pmatrix}$. Define $J_{2n} = \underbrace{\sigma \oplus \cdots \oplus \sigma}_{\text{$n$ copy}}$. We define a $2n \times 2n$ real matrix matrix $A$ as symplectic if $AJ_{2n}A^T=J_{2n}$, where $T$ is the transpose operation. I want to prove/disprove the following statement.
$Sp(2n)$ is generated by $4 \times 4$ symplectic matrices. (More explicitly, it is by matrices of the type $S$, where $X\in Sp(4)$ is a principal sub-minor, and identity on the complementary space).
Here is the sketch of a proof that I think should work:
Note that $\mathfrak{sp}(2n)$ can be defined by $$ X \in \mathfrak{sp}(2n) \iff e^{tX} \in Sp(2n) \quad \forall t \in \Bbb R $$ and given by the set of $X$ satisfying $$ J_{2n} X^T J_{2n} = X $$ this is a linear vector space. Verify that we can make a basis for this space out of the otherwise zero matrices with a principle subminor that meets this requirement.
It is a theorem that for Lie group $G$ with algebra $\mathfrak g$, the matrices $e^X$ with $X \in \mathfrak g$ generate $G$. It follows (?) that the matrices $e^X$ with $X$ in our basis generate $G$.