I find it intuitive enough that the radical of glnF is the scalar matrices, but I have trouble finding an easy, but complete proof:
Proof. Let s denote the scalar matrices. Clearly s⊂rad(glnF). Suppose that rad(glnF) is generated by more than one element, so that X∈rad(glnF), but X∉s. We can change basis such that rad(glnF) 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∈glnF s.t. [X,Y] is not upper-triangular. How can I see that Y always exists, except by waving hands?
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