I am supposed to show that there is a sequence:
$$Sp_{\mathbb{C}}(n-2) \hookrightarrow Sp_{\mathbb{C}}(n) \twoheadrightarrow M := \left\{ (x,y) \in \mathbb{C}^n \times \mathbb{C}^n \colon b(x,y) := \left\langle J \bar{x},y\right\rangle = 1 \right\} \cong TS^{2n-1} $$
where $b(\cdot,\cdot)$ is the symplectic form on $\mathbb{C}$. First of all, I have to show the isomorphism above and then prove that $Sp_{\mathbb{C}}(n)$ acts transitively on $M$ which then implies surjectivity. So far, I have no clue how to show transitivity. Any help is welcome!