A characterization of derivations and automorphisms on some simple algebras

In the present paper, we study simple algebras, which do not belong to the well-known classes of algebras (associative algebras, alternative algebras, Lie algebras, Jordan algebras, etc.). The simple finite-dimensional algebras over a field of characteristic $0$ without finite basis of identities, constructed by Kislitsin, are such algebras. In the present paper, we consider two such algebras: the simple seven-dimensional anticommutative algebra $\mathcal{D}$ and the seven-dimensional central simple commutative algebra $\mathcal{C}$. We prove that every local derivation of these algebras $\mathcal{D}$ and $\mathcal{C}$ is a derivation, and every $2$-local derivation of these algebras $\mathcal{D}$ and $\mathcal{C}$ is also a derivation. We also prove that every local automorphism of these algebras $\mathcal{D}$ and $\mathcal{C}$ is an automorphism, and every $2$-local automorphism of these algebras $\mathcal{D}$ and $\mathcal{C}$ is also an automorphism.