Proposition 5

Let $\mathfrak{g}$ be a decomposable nilpotent Lie subalgebra of $\mathfrak{g}\mathfrak{l}(V)$. Let $\mathfrak{t}$ be the set of semi-simple elements of $\mathfrak{g}$. Then $\mathfrak{t}$ is a central subalgebra of $\mathfrak{g}$, and $\mathfrak{g}$ is the product of $\mathfrak{t}$ and $\mathfrak{n}_V(\mathfrak{g})$ as Lie algebras.

If $x\in \mathfrak{t}$, ad$_{\mathfrak{g}}x$ is semi-simple and nilpotent, hence zero, so that $x$ is central in $\mathfrak{g}$. Consequently, $\mathfrak{t}$ is an ideal of $\mathfrak{g}$, and $\mathfrak{t}\cap \mathfrak{n}_V(\mathfrak{g}) = 0$. Since $\mathfrak{g}$ is decomposable, $\mathfrak{g}=\mathfrak{t}+\mathfrak{n}_V(\mathfrak{g})$, hence the proposition.