Example 1

Let $V'$ and $V''$ be vector subspaces of V such that $V''\supset V'$. The set of $x\in \mathfrak{g}\mathfrak{l}(V)$ such that $x(V'')\subset V'$ is a decomposable Lie subalgebra of $\mathfrak{g}\mathfrak{l}(V)$; indeed, for all $x\in \mathfrak{g}\mathfrak{l}$(V), the semi-simple and nilpotent components of $x$ are of the form $P(x)$ and $Q(x)$, where P and Q are polynomials without constant term.