Question

I found myself "naturally" dealing with an object of this form:

X is a complex vector space, with a "product" (a,b) → {aba} which is quadratic in the first variable, linear in the second, and satisfies some associativity conditions. These conditions are actually complicated, but more or less say that {aba} looks like the product (aba) in an alternative algebra Y containing X as a subspace.

For example, the main "associativity condition" I am interested in is: {a{b{aca}b}a}={{aba}c{aba}}

Examples

1. Symmetric matrices
2. Octonions, or indeed any alternative algebra
3. Let J belong to GL(n,ℂ), with ^tJ=-J and J²=-Id, and W={w∈M(n×n,ℂ)|J^twJ=-w}

all with the standard product {aba}=aba.

All of these examples are Jordan algebras, with respect to the symmetrized product a∘b=½(ab+ba), but I cannot see any direct link between the Jordan product and my product.