Real division algebras with a nontrivial reflection

In this note, we consider four-dimensional unital real division algebras $\mathcal A$ with $\operatorname{Aut}(\mathcal A)$ containing a nontrivial reflection $\varphi$ (i.e., an automorphism of order two). If such an algebra $\mathcal A$ is a $\mathbb C$-bimodule, then we describe its multiplication table and state division conditions in terms of certain polynomials. Finally, we suggest a new method (different from the duplication process) that can be used to construct families of four-dimensional division algebras $\mathcal A$ with $\mathfrak{Der} (\mathcal A) =\{0\}$, which are generally not third power-associative or quadratic. Under some restrictions on algebra coefficients, we have listed all possible types of their automorphism groups.