  Let $V$ be an infinite-dimensional vector space over a field. In a previous
article, we have shown that every endomorphism of $V$ splits into the sum of
four square-zero ones but also into the sum of four idempotent ones. Here, we
study decompositions into sums of three endomorphisms with prescribed split
annihilating polynomials with degree $2$. Except for endomorphisms that are the
sum of a scalar multiple of the identity and of a finite-rank endomorphism, we
achieve a simple characterization of such sums. In particular, we give a simple
characterization of the endomorphisms that split into the sum of three
square-zero ones, and we prove that every endomorphism of $V$ is a linear
combination of three idempotents.
