I find it intuitive enough that the radical of $\mathfrak{gl}_n\mathbb F$ is the scalar matrices, but I have trouble finding an easy, but complete proof:
Proof. Let $\mathfrak s$ denote the scalar matrices. Clearly $\mathfrak s\subset\mathrm{rad}(\mathfrak{gl}_n\mathbb F)$. Suppose that $\mathrm{rad}(\mathfrak{gl}_n\mathbb F)$ is generated by more than one element, so that $X\in\mathrm{rad}(\mathfrak{gl}_n\mathbb F)$, but $X\notin\mathfrak s$. We can change basis such that $\mathrm{rad}(\mathfrak{gl}_n\mathbb F)$ is upper-triangular. (The change of basis leaves the scalar matrices invariant.) Then $X$ is upper-triangular.
I want to conclude that there exists a $Y\in\mathfrak{gl}_n\mathbb F$ s.t. $[X,Y]$ is not upper-triangular. How can I see that $Y$ always exists, except by waving hands?
If $I$ is a solvable ideal of $\mathfrak{gl}_n(\mathbb{F})$ then $I\cap \mathfrak{sl}_n(\mathbb{F})$ is a solvable ideal of $\mathfrak{sl}_n(\mathbb{F})$ which is semisimple (in fact simple) as long as the characteristic of $\mathbb{F}$ does not divide $n$, so the radical of $\mathfrak{gl}_n(\mathbb{F})$ intersects trivially with $\mathfrak{sl}_n(\mathbb{F})$, and this forces it to have dimension $1$.
On the other hand, if we take the example $\mathfrak{gl}_2(\mathbb{F}_2)$ then one can check that this is actually solvable, so in this case the radical does not just consist of the scalar matrices (the derived subalgebra is contained in $\mathfrak{sl}_2(\mathbb{F}_2)$ which contains the scalar matrices. The quotient of $\mathfrak{sl}_2(\mathbb{F}_2)$ by the scalar matrices has dimension $2$ and is thus solvable).