Experience since the1970s has shown that discretization methods for low-viscosity flow problems should acknowledge the conservation laws that govern the flow. Whereas these finite-volume methods are conserving the primary invariants as described by the basic equations, during the last decades it has become clear that additional conservation of one or more secondary invariants can be very profitable, i.e. a method should be supraconservative. Especially, the attention has been focussed on discretely preserving kinetic energy: it was found that its preservation makes a discretization less sensitive to irregularities in a grid. However, this poses stringent requirements on the finite-volume discretization of most of the terms in the flow equations; only diffusion is not involved [1]. These requirements also apply to other discretization approaches such as finite-difference methods [2]. E.g., it was found that all (supra)conservative finite-difference discretizations can be written as a finite-volume method, which helps to reduce the number of approaches we have to consider. For the non-linear advective terms these requirements are all related to its so-called Picard linearization. But when a flow becomes unstable the advective Jacobian is playing a decisive role in determining its bifurcation behavior. Thus two different linearizations of the advective term are essential in flow problems. In the presentation we will study the relation between these two linearizations. In particular, with a driven-cavity flow as an example, we investigate the consequences of a supraconservative Picard discretization for the discrete Jacobian and its effects on the resulting bifurcation behavior.