Summary The article revises properties of two identification/adaptation algorithms proposed by Lion and Kreisselmeier more than 40 years ago to accelerate parametric convergence under regressor persistency of excitation (PE) condition. First, motivated by the paper3 we demonstrate that these algorithms can provide asymptotic (not exponential) parametric convergence under a simple condition which is weaker than the requirement of PE. Second, it is shown that for some special choice of adaptation gain the algorithms can provide finite time parametric convergence if the regressor satisfies the interval excitation (IE) condition that is even weaker than the condition of asymptotic convergence. Third, it is shown that for some certain structural requirements these algorithms can generate the high‐order time derivatives of the adjustable parameters. This property can be used for solution of a wide range of problems of adaptive control including, in particular, model reference adaptive control and modular backstepping design with high‐order tuners.