\(K\)-loops and quasidirect products in \(2\)-dimensional linear groups over a Pythagorean field (Q1896650)

\textit{H. Karzel} and \textit{H. Wefelscheid} [Result. Math. 23, No. 3-4, 338- 354 (1993; Zbl 0788.20034)] discussed \(K\)-loops in the Minkowski space time world over a commutative euclidean field \((K,+,\cdot)\). The author studied the problem of how far their results remain valid or how they have to be modified if one replaces the euclidean field by an ordered commutative field. In 1994 she obtained some results [ibid. 25, No. 1-2, 60-63 (1994; Zbl 0803.20051)]. Here she considers the case that \((K,+,\cdot,\leq)\) is a commutative pythagorean field. If \(L=K(i)\) is the quadratic extension with \(i^2=-1\) then she turns the future cone \({\mathfrak H}^{++}:=\{A\in\text{GL}(2,L)\mid A^*:=\overline{A}^T=A\), \(\text{det }A>0\), \(\text{Tr }A>0\}\) into a \(K\)-loop by the modified binary operation \(A\boxplus B:=\sqrt{AB^2 A}\) where \(\sqrt{A}:=(\text{Tr }A+2\sqrt{\text{det } A})^{-{1\over 2}}(\text{det }A\cdot E+A)\). If \(Q_1:=\{X\in\text{GL}(2,L)\mid X^*\cdot X=E\}\) denotes the rotation group, \({\mathfrak Q}^1:=Q_1\cap\text{SL}(2,L)\) and \({\mathfrak H}^{1+}:={\mathfrak H}^{++}\cap\text{SL}(2,L)\) then \({\mathfrak H}^{1+}\) is a \(K\)-subloop of the \(K\)-loop \(({\mathfrak H}^{++},\boxplus)\) and one can form the following quasidirect products: \({\mathfrak H}^{++}\rtimes_QQ_1\) (which is a subgroup of \(\text{GL}(2,L)\)) and \({\mathfrak H}^{1+}\rtimes_Q{\mathfrak Q}'\) (which is \(=\text{SL}(2,L)).\) If \(K\) is even euclidean then \(\text{GL}(2,L)={\mathfrak H}^{++}\rtimes_QQ_1\) and the \(K\)-loops \(({\mathfrak H}^{++},\boxplus)\) and \(({\mathfrak H}^{++},\oplus)\) with \(A\oplus B:=(\text{Tr }A+2\sqrt{\text{det }A})^{-1}(\sqrt{\text{det }A}E+A)B(\sqrt{\text{det }A}E+A)\) are isomorphic.