5 jordan algebra product

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)|JtwJ=-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.

4 associativity condition

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
3. Let J belong to GL(n,ℂ), with tJ=-J and J²=-Id, and W={w∈M(n×n,ℂ)|JtwJ=-w}

all with the standard product {aba}=aba.

All of these examples are Jordan algebras, but I cannot see any direct link between the Jordan product and my product.

3 changed ex. 2, clarifications

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.

Examples

1. Symmetric matrices
2. 3×3 Octonion matrices
3. Octonions
4. Let J belong to GL(n,ℂ), with tJ=-J and J²=-Id, and W={w∈M(n×n,ℂ)|JtwJ=-w}

all with the standard product {aba}=aba.

All of these examples are Jordan algebras, but I cannot see any direct link between the Jordan product and my product.

2 added 77 characters in body; edited tags
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