In this paper, the problem of fault estimation based on intermediate variable-based adaptive observer for a class of nonlinear systems with Lipschitzian nonlinearities and faults is studied. An adaptive observer based on lower triangular matrix transformation is proposed, which comprise the output derivatives information so as to enhance the accuracy and speed of fault estimation effectually. The designed observer can be simplified to the lower triangular estimation observer, and is superior to the comprehensive estimation observer in the related literature. In addition, a more general error dynamics is derived based on the improved adaptive estimation observer, which makes it less conservative to obtain stability conditions. The uniformly ultimately bounded stability conditions of the estimated error dynamics are given by linear matrix inequalities (LMIs). Numerical examples are given to illustrate the effectiveness and merits of the proposed method.