Jordan algebras are non-associative algebras satisfying a somewhat strange (to me) list of axioms, see wikipedia. Basic examples are real symmetric and complex hermitian matrices with the product $A\circ B=\frac{1}{2}(AB+BA)$. According to the above article in wikipedia P. Jordan introduced this notion in 1933 to formalize the notion of an algebra of observables in quantum mechanics.

**Question 1. Did this operation become really useful in quantum mechanics in any non-trivial way?**

**Question 2. I would be happy to see some explanations why the notion of Jordan algebra is useful/ natural. Are there applications of it to other parts of mathematics?**

Here are very few interesting facts I was able to find so far.

1) There is a really beautiful classification of the finite dimensional formally real Jordan algebras due to Jordan, von Neumann & Wigner (1934).

2) The above wikipedia paper mentions a classification of some special class of infinite dimensional Jordan algebras (Zelmanov, 1979).

3) I have recently heard about the Koecher-Vinberg classification of symmetric cones: such cones are precisely cones of squares in Euclidean Jordan algebras.

Thus the only way I heard Jordan algebras are related to other parts of mathematics is via the classification lists of various subclasses of them.