On diagonal nonconstant right-symmetric algebras of matrix type $M_2(F)$

We describe the right-symmetric algebras of matrix type $M_2(F)$ over a field $F$ of characteristic $0$ such that the left action of the orthogonal idempotents of $M_2(F)$ is diagonalizable, and the right-module part $W$ includes no constant bichains. We construct some wide class of nonassociative algebras $E_{\psi,\partial}(W,{\mathcal A})$, where $W$ is a subalgebra and a right module over an associative algebra ${\mathcal A}$. We give a criterion for these algebras to be right-symmetric. Assuming that $W{\mathcal A}=W$, we show that the algebras of this class are either simple or local. We exhibit some examples of simple right-symmetric algebras and right-symmetric algebras without nilpotent right ideals whose right-module part is not an irreducible module over $M_2(F)$.