Suppose that $\mathfrak{g}$ is a Lie algebra. We call $\mathfrak{g}$ a compact Lie algebra if there exists an inner product $\langle\cdot,\cdot\rangle$ on $\mathfrak{g}$ such that $$\langle[Z,X],Y\rangle+\langle X,[Z,Y]\rangle=0,\forall X,Y,Z\in\mathfrak{g}.$$On the other hand, an element $X\in\mathfrak{g}$ is called semisimple if $ad(X)$ is diagonalizable over $\mathbb{C}$, where $ad$ is the adjoint representation.
I want to show that each element in a compact Lie algebra is semisimple.