Hyperbolic loops over octaves and \(K\)-loops (Q1329596)

Let \(K\) be a euclidean field, \(L = K(i)\) the complex numbers over \(K\), \(H\) the skewfield of quaternions over \(K\) and \(O\) the alternative algebra of Cayley numbers over \(K\). We use the following notations for Hermitian matrices over \(O\) resp. \(H\) resp. \(L\): \[ \begin{multlined}{\mathfrak U} := \left\{\left( \begin{smallmatrix} \alpha & a\\ \overline{a} & \beta\end{smallmatrix} \right)\mid\alpha,\beta \in K, \;a \in O\right\},\quad {\mathfrak A} := \left\{\left( \begin{smallmatrix} \alpha & a \\ \overline{a} & \beta \end{smallmatrix} \right)\mid\alpha, \beta\in K,\;a \in H\right\},\\ {\mathfrak H} := \left\{ \left( \begin{smallmatrix} \alpha & a \\ \overline{a} & \beta \end{smallmatrix} \right)\mid\alpha, \beta \in K,\;a \in L\right\},\qquad\\ {\mathfrak U}^{++} := \{U \in {\mathfrak U} \mid \text{det }U > 0, \text{tr }U > 0\},\quad{\mathfrak U}^{1+} := \{U \in {\mathfrak U} \mid \text{det }U = 1,\;\text{tr }U > 0\}.\end{multlined} \] Analogously \({\mathfrak A}^{++}\), \({\mathfrak A}^{1+}\), \({\mathfrak H}^{++}\), \({\mathfrak H}^{1+}\) are defined. Then it is shown that all these sets of matrices are loops with respect to the operation: \[ A \oplus B := \sqrt{A} B \sqrt{A} \text{ with }\sqrt{A} = {\sqrt{\text{det}A} E + A\over \sqrt{\text{tr }A + 2\sqrt{\text{det }A}}} \] where juxtaposition means the usual matrixmultiplication. The main result of this paper consists in proving that \(({\mathfrak A}^{++},\oplus)\) and \(({\mathfrak A}^{1+},\oplus)\) are \(K\)- loops. This generalizes the earlier known result that \(({\mathfrak H}^{++},\oplus)\) and \(({\mathfrak H}^{1+},\oplus)\) are \(K\)-loops.