# Matrix Lie algebra: Semisimple element = semisimple matrix?

I am considering the set $$\mathbb{L} = \lbrace X \in \mathbb{C}^{2n \times 2n} \; ; \; J^{-1}X^HJ = -X \rbrace, \quad J = \begin{bmatrix} 0 & I_n \\ -I_n & 0 \end{bmatrix},$$ of complex Hamiltonian matrices (where superscript $$H$$ means conjugate transposition). This is a $$\mathbb{R}$$-linear subspace of matrices and, in case I'm not mistaken, an $$\mathbb{R}$$-Lie-algebra with Lie bracket $$[X,Y]=XY-YX$$ (unfortunately it is not a complex Lie algebra).

My primary research area is matrix computations and I'm not very familiar with Lie algebras. What I found out so far is:

• As a real Lie algebra of complex matrices, $$\mathbb{L}$$ is reductive (According to A. Knapp, Lie Groups - Beyond an Introduction, Prop. 1.56., $$\mathbb{L}$$ is reductive if it is closed under the operation conjugate transpose - which is the case).
• The real Lie algebra $$\mathbb{L}$$ is semisimple (again according to A. Knapp, Lie Groups - Beyond an Introduction, $$\mathbb{L}$$ is semisimple as a real Lie algebra if and only if its center is 0 - which is the case).

Is there a relation between (1) $$X \in \mathbb{L}$$ being semisimple as an element of the $$\mathbb{R}$$-Lie-algebra $$\mathbb{L}$$ and (2) $$X$$ being semisimple as a complex matrix (in the classical sense of being diagonalizable over $$\mathbb{C}$$)? In particular, does (1) imply (2)? I'm a little confused about the definition of semisimplicity in Lie algebras (e.g. via the adjoint action) and I was wondering whether these concepts in the sense (1) and (2) above are actually the same for matrices in $$\mathbb{L}$$? I would very much appreciate any help or any references.