The purpose of this paper is to revisit two problems discussed previously in the literature, both related to \(P_1P_2 = P_2P_1\), where \(P_1\) and \(P_2\) denote projectors. The first problem was considered by Baksary et al. who have shown that if \(P_1\) and \(P_2\) are orthogonal projectors, then in all nontrivial cases a product of any length having \(P_1\) and \(P_2\) as its factors occuring alternately is equal to another such product if and only if \(P_1\) and \(P_2\) commute. In the present paper a generalization of this result is proposed and validity of the equivalence between commutativity property and any equality involving two linear combinations of two any length products having orthogonal projectors \(P_1\) and \(P_2\) as their factors occurring alternately is investigated. The second problem discussed in this paper concerns specific generalized inverses of \(P_1 + P_2\) and \(P_1 - P_2\) of commuting projectors \(P_1\) and \(P_2\).