the symplectic Lie algebra

Question

I have a question about the symplectic Lie Algebra. The symplectic Lie algebra is defined as follows.

We define a skew symmetrical matrix: $S:=$ $\left( \begin{array}{rrrr} 0 & I_n \\ -I_n & 0\\ \end{array}\right) $.

Then the symplectic Lie algebra is the set $\mathfrak{sp}_{2n}:=\{A \in \mathbb{K}^{2n,2n} \mid A^TS=-SA\}$

Can one say that the symplectic Lie algebra consists of alls skew symmetrical matrices A which commute with S?

Answer 1

$$ A=\begin{pmatrix} B & 0\\ 0 & -B^T\\ \end{pmatrix}\\ A^T S = \begin{pmatrix} B^T & 0\\ 0 & -B\\ \end{pmatrix} \begin{pmatrix} 0 & I\\ -I & 0\\ \end{pmatrix}\\ = \begin{pmatrix} 0&B^T\\ B&0\\ \end{pmatrix}\\ SA= \begin{pmatrix} 0 & I\\ -I & 0\\ \end{pmatrix}\begin{pmatrix} B & 0\\ 0 & -B^T\\ \end{pmatrix}\\ = \begin{pmatrix} 0&-B^T\\ -B&0\\ \end{pmatrix} $$

So this $A$ is in the Lie algebra for all $B$. $A$ is not necessarily skew symmetric in this case because $B$ can be anything that is n by n.

Skew symmetric matrices that commute with $S$ form part of the Lie algebra, but there is more.