Abstract
In this paper we consider a four-dimensional real algebra [Formula: see text] and prove that if the quaternion group acts on a certain subset of End[Formula: see text] transitively, then [Formula: see text] is a division algebra. We will also show that under certain technical conditions [Formula: see text] has no identity. Using these results we can explicitly construct a 16-parameter family of four-dimensional real division algebras. In addition, we will find a one-parameter family of such algebras with trivial derivation algebras.
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