Control allocation (CA) is part of a hierarchical control architecture and distributes the desired control effort of a controller among a set of redundant actuators as it is used, for example, to meet high reliability requirements. Before applying CA, a factorization of a linear plant's input matrix must be carried out. This paper investigates the influence of this factorization on two common algorithms for constrained CA: direct allocation and redistributed pseudoinverse (RPINV). It is shown why the factorization does not affect the first method, whereas it influences the latter one as soon as the number of actuator saturations exceeds a certain threshold. On this basis a modification of RPINV is proposed, which allows the prioritization of virtual control vector components. If the actuator saturations prevent an exact solution, the errors of those components with high priority are preferentially forced to zero. Finally simulations demonstrate the main results.