Let $\dim(V)<\infty$ . Do there exist endomorphisms of $V$ satisfying the condition that $σ_1 + αβ − βα$ is nilpotent?

Question

Let $V$ be a vector space finitely-generated over $\mathbb{C}$. Do there exist endomorphisms $α$ and $β$ of $V$ satisfying the condition that $σ_1 + αβ − βα$ is nilpotent? (Exercise 771 from Golan, The Linear Algebra a Beginning Graduate Student Ought to Know)

I just know that if $V=\mathbb{C}$ the assumption is not true because $α$ and $β$ commute and $σ_1$ is not nilpotent, but I dont know how to star if $\dim(V)>1$.

$σ_1$ is the identity function ($σ_1:x\mapsto x$).

Thanks.

Answer 1

It is impossible to find such endomorphisms because the eigenvalues of $\alpha\beta-\beta\alpha$ would be all $-1$ in order for $\sigma_1+\alpha\beta-\beta\alpha$ to be nilpotent, a contradiction as the trace of $\alpha\beta-\beta\alpha$, which is the sum of eigenvalues, should be $0$.