Let us take matrices of the form
\begin{bmatrix} a & b\\ -b & a \end{bmatrix}
with $a,b\in \mathbb{R}$ not both zero. The determinant $a^2+b^2 \neq 0$. These matrices together with matrix multiplication, form an infinite abelian group. In Herstein's Topics in Algebra, he mentions that we should compute the product in this group and write it in the form $aI+bJ$ with $I$ the identity and $J$ being
\begin{bmatrix} 0 & 1\\ -1 & 0 \end{bmatrix}
and see if we are reminded of anything. Computing the product gives
\begin{bmatrix} ac-bd & ad+bc\\ -(ad+bc) & ac-bd \end{bmatrix}
which can be written as $(ac-bd)I+(ad+bc)J$ and clearly, this is reminiscent of complex number multiplication if we consider multiplying $a+bi$ and $c+di$ as well as their conjugates. But, I'm wondering if there is more to know about this group. I assume I've recognized what Herstein wanted us to recognize but, does this group have a special name and perhaps a special application somewhere? And what significance does it have for complex analysis, if any?
