The author proves that operators on a separable infinite-dimensional Hilbert space \(H\) of the form \((2\pm \frac{2}n)I+K\), where \(K\) is a compact operator and \(n>1\) is an integer, are decomposable into a sum of four idempotents if there exists a decomposition \(H=H_1\oplus H_1\oplus \cdots \oplus H_1\), \(K=K_1\oplus K_2\oplus \cdots \oplus K_n\), where \(K_1,\ldots K_n\) are operators on the Hilbert space \(H_1\) such that \(K_1+\cdots +K_n=0\). A decomposition of a compact operator \(K\) or the operator \(4I+K\) into a sum of four idempotents can exist only if \(K\) is finite-dimensional. On the other hand, if \(K\) is finite-dimensional and \(n\cdot \operatorname{tr}K\) is a sufficiently large (small) integer, then the operator \((2-\frac{2}n)I+K\) (resp., \((2+\frac{2}n)I+K\)) is a sum of four idempotents. Note that any operator, different from \(\lambda I+K\), \(\lambda \in \mathbb C\), is a sum of four idempotents (see \textit{C. Pearcy} and \textit{D. Topping} [Mich. Math. J. 14, 453--465 (1967; Zbl 0156.38102)]).