Jordan algebra

Remarks

Example: for an associative $F$-algebra $A$, put $x \circ y = \frac{1}{2}(xy + yx)$. Clause 1 is clear, and clause 2 follows by expanding: with $x^2 = xx$ in $A$, one has $x \circ x = x^2$, and both $(x \circ y) \circ x^2$ and $x \circ (y \circ x^2)$ equal $\frac{1}{4}(xyx^2 + yx^3 + x^3y + x^2yx)$. The resulting Jordan algebra is written $A^+$; the Hermitian $n \times n$ complex matrices form a real Jordan subalgebra of $M_n(\C)^+$. Non-example: $\R^3$ with the cross product satisfies clause 2, since $x \times x = 0$ makes both sides $0$, but fails clause 1, because $x \times y = -y \times x$. A Jordan algebra is power-associative: the powers of a single element do not depend on the bracketing.