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Most algebraic methods for Independent Component Analysis (ICA) consist of a second-order and a higher-order stage. The former can be considered as a classical Principal Component Analysis (PCA), with a three-fold goal: (a) reduction of the parameter set of unknowns to the manifold of orthogonal matrices, (b) standardization of the unknown source signals to mutually uncorrelated unit-variance signals, and (c) determination of the number of sources. In the higher-order stage the remaining unknown orthogonal factor is determined by imposing statistical independence on the source estimates. Like all correlation-based techniques, this set-up has the disadvantage that it is affected by additive Gaussian noise. However it is possible to solve the problem, in a way that is conceptually blind to additive Gaussian noise, by resorting only to higher-order cumulants. The purpose of this paper is to explain how the dimensionality of the ICA-model can algebraically be reduced to the true number of sources in higher-order-only schemes.