On nonconstant pre-Lie bimodules over $M_2(F)$

We describe the unital finite-dimensional simple nonconstant bimodules $\mathcal{W}$ over the matrix algebra $M_2(F)$ over a field $F$ of characteristic $0$; i.e., the left action of the idempotents of $M_2(F)$ is diagonalizable and $\mathcal{W}$ does not contain constant bichains. Also, we construct an example of a nondiagonal bimodule and a series of constant right-symmetric bimodules over $M_2(F)$.