Algebras whose multiplication need not satisfy the associative law (ab)c = a(bc). Three canonical non-Lie strands: Jordan algebras (commutative, satisfying the Jordan axiom (a²∘b)∘a = a²∘(b∘a)) — arose in Jordan-von Neumann-Wigner 1934 as…
nonassociative-algebra
Jordan algebra (commutative nonassociative with Jordan identity)
Jordan-von Neumann-Wigner 1934: a Jordan algebra is a commutative nonassociative R-algebra (a∘b = b∘a) satisfying the Jordan identity…
Alternative algebra (weakening of associativity: left/right alt)
An alternative algebra is a nonassociative R-algebra satisfying the left-alternative a(ab) = (aa)b and right-alternative (ab)b = a(bb)…
Flexible algebra and Malcev algebras (Jacobi generalisation)
Flexible algebras satisfy (xy)x = x(yx) — the associator is alternating in outer arguments. Both Jordan and alternative algebras are…
Sym_2(R) Jordan product: commutativity and Jordan identity literal-0 residuals
Exact symbolic verification on Sym_2(R), the 3-parameter space of 2×2 real symmetric matrices with Jordan product A∘B = (AB + BA)/2. …
Octonion associator [e1,e2,e4] = (e1·e2)·e4 − e1·(e2·e4) = 2·e7
Explicit non-associativity witness in the Cayley-Dickson octonions O on the Fano-plane multiplication convention with triples…
Jordan power-associativity: (X²)∘X = X∘(X²) = X³ on Sym_2(R)
On Sym_2(R) (symmetric 2×2 real matrices with Jordan product A∘B = (AB + BA)/2) the power-associativity identity X²∘X = X∘X² reduces to 0…