By Euler L.

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Algebra A = −a,−b F Conversely if q < Sim(q) then q does arise as the norm form of some “composition algebra” A. Any subspace σ < Sim(q) can be viewed as coming from a “partial multiplication” S × V → V . 9 Proposition. Suppose (V , q) and (S, σ ) are quadratic spaces over F . The following conditions are equivalent: (1) σ < Sim(q). (2) There is a bilinear pairing ∗ : S × V → V satisfying q(f ∗ v) = σ (f )q(v) for every f ∈ S and v ∈ V . (3) There is a formula σ (X)q(Y ) = q(Z) where each zk is a bilinear form in the systems of indeterminates X, Y with coefficients in F .

1. 6. H and H i are orthogonal, H = H + H i is a subspace on which the form is regular and dim H = 2 · dim H . Moreover, if a, b, c, d ∈ H then: ¯ + (da + bc)i. (a + bi) · (c + di) = (ac − α db) ¯ Consequently H is also a composition subalgebra of A. ¯ i] = 0. Proof. H is invariant under “bar” since 1 ∈ H . If a, b ∈ H then [a, bi] = [ba, Then H and H i are orthogonal and H ∩ H i = {0} since the form is regular. To verify the formula for products it suffices to analyze three cases. (1) [bi · c, t] = [bi, t c] ¯ = −[bc, ¯ ti] = [bc¯ · i, t].

That is, xy · z = x · yz whenever one of the factors x, y, z is equal to g. Then N (A) is an associative subalgebra of A. (1) If A is alternative it is enough to require g · xy = gx · y for every x, y. (2) If A is an octonion algebra over F then N (A) = F . (3) Does this hold true for all the Cayley–Dickson algebras An when n > 3? 36 1. Spaces of Similarities 28. Suppose A is an octonion algebra and a, b ∈ A. Then: (a, b, x) = 0 for every x ∈ A iff 1, a, b are linearly dependent. (Hint. ( ⇒) a, b ∈ H for some quaternion subalgebra H .