  We consider the possibility of developing a Lieb-Robinson bound for the
double bracket flow. This is a differential equation $$\partial_B
H(B)=[[V,H(B)],H(B)]$$ which may be used to diagonalize Hamiltonians. Here, $V$
is fixed and $H(0)=H$. We argue (but do not prove) that $H(B)$ need not
converge to a limit for nonzero real $B$ in the infinite volume limit, even
assuming several conditions on $H(0)$. However, we prove Lieb-Robinson bounds
for all $B$ for the double-bracket flow for free fermion systems, but the range
increases \emph{exponentially} with the control parameter $B$.
