Proposition 7

Suppose that A is a maximal ordered field and that $\Phi$ is a positive form (§ 7). Then the canonical homomorphisms of $O^+$ into $S^+/H^+$ and of $O^+/\{ 1, -1 \}$ into $S^+/H$ are bijective, and $S^+$ is isomorphic to $O^+ \times H^+$.

We have already seen that the homomorphisms in question are injective, and it is enough to prove that the first is surjective. Let $(e_1, e_2)$ be an orthonormal basis of E, and let $\omega$ be the direct similarity such that $\omega(e_1) = e_2$ (cor. 1 of prop. 1, No. 1); one then has $\omega^2 = -1$ (prop. 1, b)). Given an arbitrary direct similarity u = a + bw (a \in A,\ b \in A),\ \text{one has}\ N(u) = a^2 + b^2 > 0,\ \text{and there exists one and only one rotation contained in the same half-line of } A(\Phi) \text{ as } u, \text{ namely } (a^2 + b^2)^{1/2}\ u.\ \text{QED.}