Let $D$ be an indefinite quaternion algebra over $\Bbb Q$. We have a chosen isomorphism $\iota \colon D \otimes_{\Bbb Q} {\Bbb R} \cong {\mathrm{M}}_2({\Bbb R})$.
Q: If we choose another isomorphism $\iota' \colon D \otimes_{\Bbb Q} {\Bbb R} \cong {\mathrm{M}}_2({\Bbb R})$. Is it true that $\iota(D) = a\,\iota'(D)\,a^{-1}$ always for some element $a \in {\mathrm{GL}}_2({\Bbb R})$?
I do not see how I should use Skolem-Noether theorem. Please help.