In Fulton & Harris, Proposition 23.1 (computing the fundamental groups of classical complex Lie groups) and the Exercises after it, specifically in dealing with $\mathrm{Sp}_{2n}(\mathbb{C})$,
the book seems to suggest that the following two submanifolds are diffeomorphic to each other:
$$
M = \left\{ ((x_1,x_2),(y_1,y_2)) \in \mathbb{C}^{2n}\times\mathbb{C}^{2n}
: x_1^Ty_2 – x_2^Ty_1 = 1 \right\} ,
\\
M' = \left\{ ((x_1,x_2),(y_1,y_2)) \in \mathbb{C}^{2n}\times\mathbb{C}^{2n}
: x_1^Tx_1 + x_2^Tx_2 + y_1^Ty_1 + y_2^Ty_2 = 1 \right\} .
$$
In the book these are written as
$$
M = \{ (v,w) \in \mathbb{C}^{2n}\times \mathbb{C}^{2n} : Q(v,w) = 1\},
\\
M' = \{ z \in \mathbb{C}^{4n} : z^T z = 1\} ,
$$
where $Q$ is an alternating non-degenerate quadratic form on $\mathbb{C}^{2n} \times \mathbb{C}^{2n}$, which I've put in "standard form" by picking a symplectic basis.
My guess is, $M$ and $M'$ are shown diffeomorphic by showing there is a change-of-basis on $\mathbb{C}^{4n}$ which takes the
$M$ expression to the $M'$ one.
However, I'm not sure how to find such a change-of-basis; how do I complete the argument here that $M$ and $M'$ are diffeomorphic?
Perhaps I'm missing something? Any help would be much appreciated!
Edit: as Stephen has pointed out in the comments, $(v,w) \mapsto v^Tv + w^Tw$ is not at all a bilinear form (as I'd incorrectly written earlier).
Best Answer
Isn't this just, you say that $Q$ is a symplectic form on $\mathbb{C}^{2n}$, and use that there exists a basis (sometimes called (standard) symplectic basis) in which the symplectic form has the matrix representation $$ \begin{pmatrix}0_n&-1_n\\1_n&0_n\end{pmatrix}? $$ So I don't see any problem with your argument.