Let \(X\) be an infinite-dimensional real or complex Banach space, and \({\mathcal I}(X)\) be the set of all idempotents in the algebra \(B(X)\) of all bounded linear operators on \(X\). In [Stud. Math. 169, No. 1, 21--44 (2005; Zbl 1088.47030)], \textit{P. Šemrl} characterized bijective maps on \({\mathcal I}(X)\) preserving commutativity in both directions. He also described poset automorphisms of \({\mathcal I}(X)\) and bijective maps on \({\mathcal I}(X)\) preserving orthogonality in both directions. Let \(H\) be an infinite-dimensional real or complex Hilbert space and \({\mathcal I}_{\infty}(H)\) be the set of all bounded linear idempotent operators on \(H\) for which both the range and kernel have infinite dimensions. Motivated by Šemrl's results, the author of the paper under review characterized three types of maps mentioned above but on \({\mathcal I}_{\infty}(H)\) instead of \({\mathcal I}(X)\). He restricts his studies to the Hilbert space case because there are reflexive Banach spaces \(X\) for which \({\mathcal I}_{\infty}(X)\) is empty as they cannot be written as direct sum of two infinite-dimensional closed linear subspaces.