The idea of using estimation algebra to construct finite-dimensional nonlinear filters was first proposed independently by Brockett and Mitter. Estimation algebra turns out to be a useful concept in the investigation of finite-dimensional nonlinear filters. In his talk at the International Congress of Mathematics in 1983, Brockett proposed classifying all finite-dimensional estimation algebras, Chiou and the present authors classify all finite-dimensional estimation algebras of maximal rank with dimension of the state space less than or equal to three. In this paper, we succeed in classifying all finite-dimensional estimation algebras of maximal rank with state-space dimension equal to four, In fact our method gives classification of all finite-dimensional algebras of maximal rank with state-space dimension equal to or less than four.
