Four-dimensional real division algebras with few derivations

In order to advance in the determination of all four-dimensional real division algebras $\mathcal{A}$, we introduce a new duplication process preserving the unit. This duplication process accompanied by an isotopy allow us to obtain all of these algebras in case $\operatorname{Der}(\mathcal{A})\neq\{0\}$ and partially in case $\operatorname{Der}(\mathcal{A})=\{0\}$. In the last case, we provide an $8$-parameter family of ugly $\mathbb C$-associative algebras and an $8$-parameter family of $\mathbb C$-associative algebras whose group of automorphisms contains only the identity and some reflections. For non-ugly algebras $\mathbb H_{f, f}$, the group $\operatorname{Aut}(\mathbb H_{f, f})$ contains a reflection. Also algebras $\mathcal A$ with $\operatorname{Aut}(\mathcal A)=\mathbb Z_2$ or $\mathbb Z_2\times\mathbb Z_2$ are studied.