Let $\mathfrak{g} = \{X ∈ gl_{2n} (\Bbb R), ^tXJ + JX = 0\}$ where $J=\begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix}$
Let $i ∈ [1, ...,n]$ and $H_i = E_{i,i} − E_{n+i,n+i} (i = 1, ..., n)$.
Prove that $\exists f\in Aut(\mathfrak{g})$ such that $f (H_i ) = −H_i$ and $f (H_j ) = H_j$ for $i ∈ [1, ...,n]\setminus \{i\}$
Prove that $B(H_i , H_j ) = 0$
What I did:
I found that $\{E_{k,l, k\ne l}, H_{i,i\in \{1,...,n\}}\}$ is a basis for $\mathfrak{g}$. so an example of such $f\in Aut(\mathfrak{g})$ could be:
$f: H_i\to -H_i$
$H_j\to H_j$ $(j\ne i)$
$E_{k,l, k\ne l} \to E_{k,l, k\ne l}$
This is an automorphism as it sends a basis of $\mathfrak{g}$ to a basis of $\mathfrak{g}$.
Now concerning $B(H_i , H_j ) = 0$, I have no clue on how to proceed.
Could you please give me help or hints? Thank you.