On nil-elements and idempotents in the simple three-dimensional commutative unital algebras

We study simple $3$-dimensional commutative unital algebras containing nil-elements of index $3$ in which nil-elements of index $2$ are absent. It is proved that each such algebra depends on either one or two parameters. For each of the parametric algebras we find their idempotents and nil-elements under some natural restrictions on the parameters. In addition, we study idempotents and nil-elements in one isotope of some special algebra.