Is every endomorphism a linear combination of projections?

Question

Let $E$ be a vector space over a field $K$. If $u \in \mathscr L(E)$ is an endomorphism of $E$, can it be written as a linear combination of projections (i.e. endomorphisms $p$ of $E$ such that $p \circ p = p$)?

If the dimension of $E$ is finite, this is true. Indeed, one can check that if $(E_{i,j})_{1 \leq i,j \leq n}$ is the canonical basis of $\mathscr M_n(K)$, then $B = (E_{i,i})_{1 \leq i \leq n} \cup (E_{i,i}+E_{i,j})_{1 \leq i,j \leq n,\,i \neq j}$ is a basis of $\mathscr M_n(K)$ and every element of $B$ is a projection matrix.

If $E$ is an Hilbert space and $u$ is bounded, this is also true (see for instance Fillmore, Peter A. "Sums of operators with square zero." Acta Scientiarum Mathematicarum 28.3-4 (1967): 285).

In the general case (i.e. arbitrary vector spaces $E$ of infinite dimension), does it still hold ?

Answer 1

The answer is yes: see Goerge Lowther's answer at Pierre-Yves Gaillard's MathSE linked question: every endomorphism is a $K$-linear combination of $\le 9$ idempotents.

In infinite dimension, the argument even shows that every endomorphism is a $\mathbf{Z}$-linear combination of idempotents (i.e., a sum of $\pm$ idempotents). In finite dimension however this is not true: an endomorphism is a $\mathbf{Z}$-linear combination of idempotents iff its trace belongs to $\mathbf{Z}1_K$.