Let $\sigma$ be a Dynkin automorphism of $G=\mathrm{SO}_{2n}$. By definition, $\sigma$ stabilizes a maximal torus $T$ and a Borel subgroup $B$, and preserves a pinning of $G$. My question is how to define this map $\sigma$ explicitly. I have tried
$\sigma:A \mapsto XAX$
where $$X=
\begin{pmatrix}
-1 & 0 & 0 & \cdots & 0\\
0 & 1 & 0 & \cdots & 0\\
0 & 0 & 1 & \cdots & 0\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
0 & 0 & 0 & \cdots & 1
\end{pmatrix}.
$$
Is this $\sigma$ a Dynkin automorphism of $G$? Are there other ways to define such a map $\sigma$?