Compact symplectic Lie algebra ($\mathfrak{usp}(2n)\cong \mathfrak{sp}(n)$)

The Lie algebra of the compact symplectic group consists of $X^{†}=-X$ and $XJ+J{X}^T=0$. Equivalently $X=\left(\begin{array}{cc}A& B\\ -\overline{B}& \overline{A}\end{array}\right)$ with $A^{†}=-A$ and $B^T=B$. Its real dimension is $n(2n+1)$, and its complexification is the symplectic Lie algebra.

In its block form, a matrix-unit basis of the unitary Lie algebra spans the anti-Hermitian block $A$, with the corresponding conjugate block below. Independent real and imaginary symmetric matrix units span $B$, with lower block $-\overline{B}$. These give $n^2+n(n+1)$ independent real generators, including diagonal symmetric entries.