sp(4,R) — real symplectic Lie algebra

The real symplectic Lie algebra $\mathfrak{sp}(4,\mathbb R)$ consists of matrices preserving the standard alternating form infinitesimally:

\mathfrak{sp}(4,\mathbb R)=\{X\in M_{4}(\mathbb R):X^T J+JX=0\},\qquad J=\begin{pmatrix}0&I_{2}\\-I_{2}&0\end{pmatrix}.

The bracket is $[X,Y]=XY-YX$ . Transpose is used without conjugation.

Block description and size convention

Writing matrices in four equal blocks, the defining equation gives

X=\begin{pmatrix}A&B\\C&-A^T\end{pmatrix},\qquad B=B^T,\quad C=C^T.

Thus there are $4+2\cdot3=10$ independent real parameters. This is the split real symplectic algebra.

The number $4$ is the full matrix size; the compact quaternionic algebra $\mathfrak{sp}(2)$ is a different real form.