Frobenius's Theorem/Lemma 3

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

Then:
 * $A = \R \oplus U$

Proof
Let $a \in A \setminus \R$.

Then:
 * $\exists \nu \in \R: a^2 - \nu a \in \R$

Therefore, if we set
 * $u = a - \dfrac \nu 2 \in U$

then $u^2 = a^2 - \nu a + \nu^2/4 \in \R$, so
 * $a = \dfrac \nu 2 + u \in \R \oplus U$

which proves the assertion.