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Abstract: Generalizing the result in Lemma of Baksalary and Baksalary [J.K. Baksalary, O.M. Baksalary, Commutativity of projectors, Linear Algebra Appl. 341 (2002) 129–142], Baksalary et al. [J.K. Baksalary, O.M. Baksalary, T. Szulc, Linear Algebra Appl. 354 (2002) 35–39] have shown that if P 1 and P 2 are orthogonal projectors, then, in all nontrivial situations, a product of any length having P 1 and P 2 as its factors occurring alternately is equal to another such product if and only if P 1 and P 2 commute, in which case all products involving P 1 and P 2 reduce to the orthogonal projector P 1 P 2 (= P 2 P 1). In the present paper, further generalizations of this property are established. They consist in replacing a product of the type specified above, appearing on the left-hand side (say) of the equality under considerations, by an affine combination of two or three such products. Comments on the problem when the number of components in a combination exceeds three are also given. [Copyright &y& Elsevier]