Quantifier Elimination and Invariant Theory: Applications to Quaternions, Octonions, and Other Algebras

We build on our previous paper \cite{constructive} by using the general method introduced there in conjunction with invariant theory. This yields quantifier elimination results for the classical quaternions, octonions, as well as other classes of finite-dimensional algebras over real closed and algebraically closed fields. In particular, the first two examples answer an open question posed recently in \cite{savi}.