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