Symplectic structures on Hermitian matrices

Question

This is a question taken from Ana Cannas da silva's book on symplectic geometry. Let $\xi\in\mathcal{H}$, the vector space of $n\times n$ hermitian matrix. Define $\omega_{\xi}(X,Y)=i\,\text{trace}([X,Y]\xi)$ where $X,Y\in i\mathcal{H}$ is the skew symmetric hermitian matrix. The author then claims that $\omega_{\xi}=i\,\text{trace}(X[Y,\xi])$. I didn't see why this is true. Can someone help me with this? Thank you very much!

Answer 1

Note that $\DeclareMathOperator{\trace}{trace}\trace(A + B) = \trace(A) + \trace(B)$ and $\trace(AB) = \trace(BA)$. It follows that $$ \trace([X,Y]\xi) =\\ \trace(XY \xi - YX \xi)=\\ \trace(XY \xi) - \trace(YX \xi)=\\ \trace(XY \xi) - \trace(Y(X \xi))=\\ \trace(XY \xi) - \trace((X \xi)Y)=\\ \trace(X(Y\xi - \xi Y)) =\\ \trace(X[Y,\xi]) $$