Frobenius's Theorem/Lemma 2

Lemma

Let $\struct {A, \oplus}$ be a quadratic real algebra.

Then:

$\forall u, v \in U: u v + v u \in \R$

Proof

From Lemma 1:

$\forall u, v \in U: u v + v u = \paren {u + v}^2 - u^2 - v^2 \in \R$.

$\blacksquare$