I'm trying to make sense of a definition appearing in Hua Luogeng's book "Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains".

Consider the Hermitian symmetric space of noncompact type
$$
D=\{z\in M_n(\mathbb C) \, | \, 1-z^*z>0 \mbox{ and } z^\text{t}=-z\}.
$$
Hua says the automorphism group of this domain is composed of matrices of the form
$$
\begin{pmatrix}
A & B \\ -\bar B & \bar A
\end{pmatrix}\in M_{2n}(\mathbb C)
$$
with
$$
A^\text{t} B=-B^\text{t}A,\quad A^*A-B^*B=\mathbb I_n,
$$
acting on $z\in D$ as
$$
(Az+B)(-\bar B z+\bar A)^{-1}.
$$
However, we know from Cartan's classification that
$$\mathrm{Aut}(D)=\mathrm{SO}^*(2n) = \{ M \in \mathrm{SU}(n,n) \,|\, M^\text{t}\gamma M = \gamma\},
\quad
\gamma=
\begin{pmatrix}
0 & \mathbb I_n \\
\mathbb I_n & 0
\end{pmatrix},
$$
i.e. consists of the matrices of the form
$$
g=
\begin{pmatrix}
A & B\\C & D
\end{pmatrix}\in M_{2n}(\mathbb C)
$$
acting on $z\in D$ as
$$
gz=(Az+B)(Cz+D)^{-1}
$$
where $\det(g)=1$ and
$$
\begin{align}
A^*A - C^*C &=\mathbb I_n \\
D^*D - B^*B &=\mathbb I_n \\
A^\text{t}D + C^\text{t} B &=\mathbb I_n
\end{align}
$$
$$
\begin{align}
A^*B &= C^*D \\
A^\text{t} C &= - C^\text{t}A \\
B^\text{t}D &= - D^\text{t}B.
\end{align}
$$
**Why does Hua's definition of the automorphism group define $C$ and $D$ repsectively as $-\bar B$ and $\bar A$?**
The book is very concise and doesn't explain what is going on, nor does it have any reference for this definition. Am I missing something? Note that although I am using $\mathrm{SO}^*(2n)$ and antisymmetric $z$, the same thing happens with respectively $Sp(n,\mathbb{R})$ and symmetric $z$, with some sign changes.

## Update #1

So far what I was able to find is that the constraints on the matrices $A, B,C,D$ guarantee that both $A$ and $D$ are invertible, and that $$ \det(g) = 1 \quad\Rightarrow \quad \det(D)=\det(\bar A). $$ This of course doesn't prove that $D=\bar A$, but at least is consistent with it. Similarly I was able to find that $$ \det(C)=(-1)^n \det(\bar B). $$