Dissident algebras

Given a euclidean vector space V = (V,〈〉) and a linear map η: V ∧ V → V, the anti-commutative algebra (V,η) is called dissident in case η(v ∧ w) ∉ ℝv ⊕ ℝw for each pair of non-proportional vectors (v,w) ∈ ${\displaystyle V^2}$. For any dissident algebra (V,η) and any linear form ξ: V ∧ V → ℝ, the vector space ℝ × V, endowed with the multiplication (α,v)(β,w) = (αβ -〈v,w〉+ ξ(v ∧ w), αw + βv + η(v ∧ w)), is a quadratic division algebra. Up to isomorphism, each real quadratic division algebra arises in this way. Vector product algebras are classical special cases of dissident algebras. Via composition with definite endomorphisms they produce new dissident algebras, thus initiating a construction of dissident algebras in all possible dimensions m ∈ {0,1,3,7} and of real quadratic division algebras in all possible dimensions n ∈ {1,2,4,8}. For m ≤ 3 and n ≤ 4, this construction yields complete classifications. For m=7 it produces a 28-parameter family of pairwise non-isomorphic dissident algebras. For n=8 it produces a 49-parameter family of pairwise non-isomorphic real quadratic division algebras.