Do matrices of this type form a group? I have a set of matrices of the form
$$A = \begin{pmatrix}
a & b\\
-b & a
\end{pmatrix}$$
where $a,b \in \mathbb{C}$. They also have the property
$$A A^{T} = (a^{2} + b^{2})\mathbb{I}$$
We do know that they form a group with two conditions:

*

*$a^{2} + b^{2} \neq 0$

*${|a|
}^{2} + {|b|}^{2} \neq 0$
What group do these matrices form?
 A: The Wikipedia article Bicomplex number
has the key to the answer. It states that

The general bicomplex number can be represented by the matrix ${w\: iz \choose iz \:w}$ which
has determinant $w^2+z^2$.

Use a different basis and the matrix becomes
${a\:\: b \choose -b \:a}$ which has
determinant $a^2+b^2$ which form a complex
algebra which is a ring because they are closed under addition and multiplication with basis
$\,\{I,J\}\,$ where
$$I^2=I,\,IJ=JI=J,\,J^2=-I.$$
One matrix representation is
$$ I={1\:0 \choose 0\:1},\quad
J={0\:1 \choose -1\:0}  $$
and another is
$$ I={1\:0 \choose 0\:1},\quad
J={0\:i \choose i\:0} . $$
The Wikipedia article also states

The bicomplex numbers form a commutative algebra over $\,\mathbb{C}\,$ of dimension two, which is isomorphic to the direct sum of algebras
$\,\mathbb{C}\oplus\mathbb{C}.$

The isomorphism is similar to the case of
split-complex numbers
$$aI+bJ\;\;\leftrightarrow\;\;(a+b\,i,a-b\,i)$$
where addition and multiplication are defined
componentwise which implies that the group of
units is $\,\mathbb C^{\times}\times\mathbb
 C^{\times}$ with all numbers of the form
$\,a\,I\pm a\,i\,J\,$ being the zero divisors.
