I'm reading the follow book: Nilpotent orbit on semisimple complex Lie Algebra . In the lemma $4.1.3$## Heading ## the autors uses the following assertion:
Let, $\{H,X,Y\}$ a $\mathfrak{sl}_2$-triple on a complex semisimple Lie Algebra $\mathfrak{g}$ and consider a decomposition of $\mathfrak{g}$ as a direct sum of irreductible $\mathfrak{a}$-modules, where $\mathfrak{a}$ is the subalgebra of $\mathfrak{g}$ generated by $\{H,X,Y\}$. Then, each of such submodule contributes a $1$-dimensional subespace to $\mathfrak{g}^X=\{Y\in \mathfrak{g}|[X,Y]=0\}$.
I don't understand why it is true. The only thing i know is that since $H$ is semisimple, this implies that $\mathfrak{g}$ has a $ad_H$-eigenspaces decomposition, where each eigenspace is associated with a integer eigenvalue and the elments of the center of $X$ are associeted with positive eigenvalues of $ad_H$.