  Let $T\in B(H)$ be such that $T^n$ is self-adjoint for some $n\in\mathbb{N}$
with $n\geq 3$. The paper's primary aim is to establish the conditions that
lead to the self-adjointness of $T$. We pay particular attention to the case
where $T^3=0$ and how it implies $T$ is complex symmetric.
