Exercise 4, § 13

Let $S$ be the square matrix of order $n$

0 0 $\cdots$ 0 1

0 0 $\cdots$ 1 0

$(\delta_{i,n+1-i}) =$ ... ... . .. ... ....

0 1 $\cdots$ 0 0

1 0 $\cdots$ 0 0

The elements of $\mathfrak{o}_S(n, k)$ are the matrices $(a_{ij})$ that are anti-symmetric with respect to the second diagonal:

$a_{i,j}=-a_{n+1-j,n+1-i}$ for every pair $(i, j)$.

The algebra $\mathfrak{o}_S(n, k)$ is splittable simple of type $D_{n/2}$ if $n$ is even and $\geq 6$, and of type $B_{(n-1)/2}$ if $n$ is odd and $\geq 5$. The diagonal (resp. upper triangular) elements of $\mathfrak{o}_S(n, k)$ form a splitting Cartan (resp. Borel) subalgebra.