I have an isospectral flow on $\mathfrak{su}(N)$, for some $N>1$ of the form: $$ \dot{W}=\left[ LW,W\right], $$ where $W\in\mathfrak{su}(N)$ and, introducing the Frobenius inner product $\langle A,B\rangle_F=Tr(A^*B)$, $L$ is an invertible positive definite Frobenius self-adjoint linear map on $\mathfrak{su}(N)$.
At $t=0$, assume that $LW$ and $W$ do not commute: $$ \left[ LW,W\right](0)\neq 0. $$ Is it possible to prove that $$ \lim_{t\rightarrow\infty}\|\left[ LW,W\right](t)\|>0 ? $$ The spectrum $\sigma(W)$ and the energy $H(W) = -1/2 Tr(LW^*W)$ are first integrals of motion.