Skew-adjoint nilpotent operators vanish
= Skew-adjoint nilpotent operators vanish
{title2=$A^*=-A,\ A^k=0\ \Longrightarrow\ A=0$}
On a finite-dimensional positive inner-product space, a skew-adjoint <nilpotent operator> is zero. In an orthonormal basis, $\|A\|_F^2=\operatorname{tr}(A^*A)=-\operatorname{tr}(A^2)=0$, since a <nilpotent matrix> has zero trace and so does its square. The <Frobenius norm> therefore vanishes. In particular, a nonzero nilpotent adjoint map rules out a <compact Lie algebra>.