Jordan, von Neumann and Wigner  showed that for any finite-dimensional real vector space $A$ with a bilinear commutative power-associative operation $\circ : A \times A \to A$, the formal reality condition
$$ a_1 \circ a_1 + \cdots + a_n \circ a_n = 0 \implies a_1, \dots, a_n = 0 $$
implies the remaining Jordan algebra axiom: for all $a \in A$, multiplication by $a \circ a$ commutes with multiplication by $a$.
Thus, in the definition of a formally real Jordan algebra (which is the kind of Jordan algebra important in quantum mechanics), we can leave out the law $(a \circ a) \circ (a \circ b) = a \circ ((a \circ a) \circ b)$ without harm, since it follows from the other conditions... in the finite-dimensional case.
But their proof uses finite-dimensionality. So, my question is:
Do you know an example of an infinite-dimensional vector space with a commutative bilinear power-associative operation obeying the formal reality condition that is not a Jordan algebra?
 P. Jordan, J. von Neumann and E. Wigner, On an algebraic generalization of the quantum mechanical formalism, Ann. Math. 35 (1934), 29-64.