I have a problem understanding a passage from "Naive Lie theory"(by John Stillwell), here is the passage from section $3.9$ ,page $71$:
The idea of treating orthogonal, unitary, and symplectic groups uniformly as generalized isometry groups of the spaces $R^n$, $C^n$, and $H^n$ seems to be due to Chevalley [1946]. Before the appearance of Chevalley’s book, the symplectic group $Sp(n)$ was generally viewed as the group of unitary transformations of $C^{2n}$ that preserve the symplectic form:
\begin{equation} (\alpha_1 \bar\alpha'{_1}-\beta_1 \bar\beta'{_1})+ ...+(\alpha_n \bar\alpha'{_n}-\beta_n \bar\beta'{_n}) \end{equation}
where $(α_1,β_1, . . . ,α_n,β_n)$ is the typical element of $C^{2n}$. This element corresponds to the element $(q_1, . . . ,q_n)$ of $H^n$, where:
\begin{equation} q_i=\begin{bmatrix} \alpha_i & -\beta_i \\ \bar\beta_i & \bar\alpha_i \end{bmatrix} \end{equation}
The invariance of the quaternion inner product:
\begin{equation} q_1 \bar{q}'{_1}+...+q_n \bar{q}'{_n} \end{equation} is therefore equivalent to the invariance of the matrix product:
\begin{equation} \left( \begin{array}{ccc} \alpha_1 & -\beta_1\\ \bar{\beta_1} & \bar{\alpha_1}\end{array} \right)\ \left( \begin{array}{ccc}\ \bar{\alpha'_1} & \beta'_1\\ -\bar{\beta'_1} & \alpha'_1\end{array} \right)+...+ \left( \begin{array}{ccc} \alpha_n & -\beta_n\\ \bar{\beta_n} & \bar{\alpha_n}\end{array} \right)\ \left( \begin{array}{ccc}\ \bar{\alpha'_n} & \beta'_n\\ -\bar{\beta'_n} & \alpha'_n\end{array} \right) \end{equation}
which turns out to be equivalent to the invariance of the symplectic form.
My problem is that i couldn't see how the invariance of matrix product results in invariance of symplectic form mentioned by the author. I have checked the Chevalley book but i didn't find the specific symplectic form used by Stillwell. Chevalley and other authors have used a symplectic form that doesn't include complex conjugate. This free complex conjugate symplectic form makes sense because it is the off-diagonal element of matrix product above, .i.e:
\begin{equation} (\alpha_1 \beta'{_1}-\beta_1 \alpha'{_1})+ ...+(\alpha_n \beta'{_n}-\beta_n \alpha'{_n}) \end{equation}
Other authors also use symplectic form without complex conjugate, for example
Modern Geometric Structures and Fields
But i don't know how to see the symplectic form defined by Stillwell is remained invariant.
What am i missing?
