$\gdef\R{\mathbf{R}}$ $\gdef\spaces#1{~ #1 ~}$

$\R[x]/(x^2+1)$

$z = a+b\sqrt{-1}$, $|z| = \sqrt{a^2+b^2}$, $\det A = a^2+b^2 = |z|^2$

$$ \left{ \begin{pmatrix} a & -b \ b & ~~~a \end{pmatrix} : a, b \in \R \right} \spaces\subset \text{GL}_2(\R) $$