Sums of three quadratic endomorphisms of an infinite-dimensional vector space

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.