restrict the “range” for a famous isomorphism

z:= x+yi
w:= a+bi

zw = (xa-yb) + (xb+ya)i

u:= xa-yb
v:= xb+ya

f: \Bbb C \rightarrow M_2(\Bbb R)
f(p+qi) := \begin{pmatrix} p & q \\ -q & p\end{pmatrix}  (p, q \in \Bbb R)

f(z)f(w) = \begin{pmatrix} x & y \\ -y & x\end{pmatrix} \begin{pmatrix} a & b \\ -b & a \end{pmatrix} =

=\begin{pmatrix} xa-yb & xb+ya \\  -ya-xb & xa-yb \end{pmatrix} = \begin{pmatrix} u & v \\ -v & u\end{pmatrix} = f(zw)

f(z)+f(w) = f(z+w) trivially.

PS

\begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix} \begin{pmatrix} \cos\psi & \sin\psi \\ -\sin\psi & \cos\psi \end{pmatrix} =

=\begin{pmatrix} \cos\theta\cos\psi - \sin\theta\sin\psi & \cos\theta\sin\psi+\sin\theta\cos\psi \\ -\cos\theta\sin\psi -\sin\theta\cos\psi & \cos\theta\cos\psi - \sin\theta\sin\psi \end{pmatrix}=

=\begin{pmatrix} \cos(\theta+\psi) & \sin(\theta+\psi) \\ -\sin(\theta+\psi) & \cos(\theta + \psi) \end{pmatrix}

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