Unital $3$-dimensional structurable algebras: classification, properties and $\rm{AK}$-construction

This paper is devoted to the classification and studying properties of complex unital $3$-dimensional structurable algebras. We provide a complete list of non-isomorphic classes, identifying five algebras for type $(2, 1)$ and two algebras for type $(1, 2).$ For each obtained algebra, we describe the derivation algebra, the automorphism group, the lattice of subalgebras and ideals, and functional identities of degree $2$. Furthermore, we investigate the Allison-Kantor construction for the classified algebras. We determine the structure of the resulting $\mathbb{Z}$-graded Lie algebras, providing their dimensions and Levi decompositions.