Quadratic Jordan algebra

Summary

A quadratic Jordan algebra is a version of a Jordan algebra built on the quadratic operators $U_x$ instead of the bilinear product. In an associative algebra, $U_x y = xyx$. The notion, due to Kevin McCrimmon, works over any commutative ring, including rings in which $2$ is not invertible. When $\frac{1}{2}$ is available it is equivalent to the usual notion of a unital Jordan algebra.