K-loops and quasidirect products in 2-dlmensional Linear Groups over a Pythagorean Field

For a group G = (G, ·), we define the (internal) quasidirect product f · U = F × U of a certain K-loop (F,+) with F ? G and a suitable subgroup il of G (cf. (3.1)). Let K be a commutative pythagorean field and let L = K(i) be the quadratic extension of K with i2 = ～-1. Then the future cone H:= A ∈ GL(2,L) ¦ A = A*, det A ∈ K⁺, Tr A ∈ K⁺ is a K-loop with respect to the binary operation $A?ggsquaredplus B:=sqrt{AB^{2}A},{? where}sqrt{A}=({? Tr}A+2sqrt {{? det}A})^{1?er 2}(sqrt {? det}AE+A)$} (cf. (2.4)), and the (internal) quasidirect product $H^{}</Emphasis>{\mathop \times\limits_{Q}}Q_{1}$ of the K-loop (H},+) and the group Q₁:= {X ∈ GL(2,L) ¦ X*X = E) is a subgroup of GL(2,L) (cf. (3.2)). Moreover, S L(2,1) = $H^{1+}{\mathop \times\limits_{Q}}Q^{1}$ , where H¹+ = SL(2,L)∩ H ≤} (H},+), Q¹ = S L(2, L) ∩ Q₁ (cf. (3.4)), and if K is euclidean, then (cf. (3.6)).