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We consider the possibility of developing a Lieb–Robinson bound for the double bracket flow (Brockett, 1991; Chu and Driessel, 1990). This is a differential equation which may be used to diagonalize Hamiltonians. Here, is fixed and . We argue (but do not prove) that need not converge to a limit for nonzero real in the infinite volume limit, even assuming several conditions on . However, we prove Lieb–Robinson bounds for all for the double bracket flow for free fermion systems, but the range increases exponentially with the control parameter .