sp(2,R) — real symplectic Lie algebra

The real symplectic Lie algebra $\mathfrak{sp}(2,\mathbb R)$ consists of matrices preserving the standard alternating form infinitesimally:

\mathfrak{sp}(2,\mathbb R)=\{X\in M_{2}(\mathbb R):X^T J+JX=0\},\qquad J=\begin{pmatrix}0&I_{1}\\-I_{1}&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 $1+2\cdot1=3$ independent real parameters. This is the split real symplectic algebra. In matrix size two the condition is exactly trace zero, giving $\mathfrak{sp}(2,\mathbb R)=\mathfrak{sl}(2,\mathbb R)$ .

The number $2$ is the full matrix size; the compact quaternionic algebra $\mathfrak{sp}(1)$ is a different real form.