A classification of 2-dimensional endo-commutative straight algebras of type II_1

In this paper, we present a complete classification of 2-dimensional endo-commutative straight algebras of type II${}_1$ over any field. An endo-commutative algebra is a non-associative algebra in which the square mapping preserves multiplication. A 2-dimensional straight algebra satisfies the condition that there exists an element $x$ such that $x$ and $x^2$ are linearly independent. The term type II${}_1$ denotes a distinguishing characteristic of its structure matrix, which has rank 2. We provide multiplication tables for these algebras, listing them up to isomorphism.