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 Open image in new window 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, Open image in new window, where H1+ = SL(2,L)∩ H ≤} (H},+), Q1= S L(2, L) ∩ Q1 (cf. (3.4)), and if K is euclidean, then Open image in new window (cf. (3.6)).