HOLGER P. PETERSSON. Meinem Bruder, Jörn P. Petersson, zur Vollendung des 80. Lebensjahres gewidmet

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1 THE NON-ORTHOGONAL CAYLEY-DICKSON CONSTRUCTION AND THE OCTONIONIC STRUCTURE OF THE E 8 -LATTICE HOLGER P. PETERSSON Meinem Bruder, Jörn P. Petersson, zur Vollendung des 80. Lebensjahres gewidmet Abstract. Using a conic = degree-2 algebra B over an arbitrary commutative ring, a scalar µ and a linear form s on B as input, the non-orthogonal Cayley-Dickson construction produces a conic algebra C := CayB; µ, s and collapses to the standard orthogonal Cayley-Dickson construction for s = 0. Conditions on B, µ, s that are necessary and sufficient for C to satisfy various algebraic properties like associativity or alternativity are derived. Sufficient conditions guaranteeing non-singularity of C even if B is singular are also given. As an application we show how the algebras of Hurwitz quaternions and of Dickson or Coxeter octonions over the rational integers can be obtained from the non-orthogonal Cayley-Dickson construction. Contents 1. Introduction 1 2. Conic algebras 2 3. Elementary properties of the non-orthogonal Cayley-Dickson construction 3 4. The conjugation, norm-associativity and flexibility 6 5. Commutativity Associativity Alternativity Non-singularity Examples: Hurwitz, Dickson and Coxeter 34 References Introduction The arithmetic background of the present paper is dominated by the E 8 -lattice, the unique indecomposable unimodular positive definite quadratic lattice of rank 8 over the integers that has recently come to the fore again when Viazovska [19 proved that the densest sphere packing of eight-dimensional euclidean space is the E 8 -lattice sphere packing, having density π 4 /384. Our aim here is much more modest, focusing instead on a discovery attributed to Coxeter [2 see also Pumplün [17, but originally due to Dickson [3, to the effect that the E 8 -lattice carries the structure of an octonion algebra over the integers whose generic fiber is the unique octonion division algebra over the rationals. Working over an arbitrary commutative associative ring of scalars, our principal objective in this paper will be to describe an elementary, purely algebraic formalism, called the non-orthogonal Cayley-Dickson construction, that, among other things, provides an intrinisc approach to the Dickson and Coxeter octonions once the appropriate specifications have been made, see 9 below for details. This formalism, generalizing the classical Cayley-Dickson construction, one of the most versatile tools in all of non-associative algebra, has been investigated before by Garibaldi-Petersson [5, 4 under the more restrictive condition of a base field having characteristic Mathematics Subject Classification. Primary 17A45; Secondary 17A75, 11E12. 1

2 2 HOLGER P. PETERSSON The classical Cayley-Dickson construction starts from what we call a conic algebra B see 2.1 for the precise definition and a scalar µ in the base ring as input to produce a new conic algebra as output, which we denote by C := CayB, µ. One of the many interesting features of this construction is that, by passing from B to C we are bound to lose a considerable amount of algebraic information, but we will do so in a controlled manner. The usefulness of its non-orthogonal counterpart hinges on the question of whether the same amount of control can be guaranteed also under these more general circumstances. The bulk of the present work is devoted to answering this question in the affirmative. Unfortunately, we are able to achieve this objective only by an excessive amount of horrendous computations. More specifically, the input of the non-orthogonal Cayley-Dickson construction, beside the data B, µ as above, consists of a linear form s acting on B, while the output is again a conic algebra, denoted by C := CayB; µ, s. The main task we address ourselves to will then be to find conditions in terms of B, µ, s that are necessary and sufficient for the algebra C to be respectively commutative, associative or alternative. While the restrictions on B and µ are to be expected from the orthogonal case, it is the ones on s that make our investigation delicate and cumbersome. We refer to 4 7 for details. Throughout this paper, we fix an arbitrary commutative ring denoted by k. All k- algebras are assumed to be non-associative; their module structure is arbitrary 2. Conic algebras In this section, we define the notion of a conic algebra and recall some of its most useful properties. Our main reference is [ The concept of a conic algebra. Adopting the terminology of Loos [10, we define a conic algebra over k, more commonly known under the name algebra of degree 2 McCrimmon [12 or quadratic algebra Osborn [15, as a unital k-algebra C together with a quadratic form n C : C k, by abuse of language called the norm of C, such that n C 1 C = 1 and x 2 t C xx + n C x1 C = 0 for all x C. Here t C : C k is the trace of C defined as the linear form x n C 1 C, x, where x, y n C x, y := n C x + y n C x n C y stands for the bilinearization of n C. We then define the conjugation of C as the map ι C : C C, x x := t C x1 C x, which is linear of period 2 but will fail in general to be an algebra involution Basic identities. The following identities by [16, 18.5 hold in arbitrary conic algebras x 2 = t C xx n C x1 C, t C x = n C 1 C, x, n C 1 C = 1, t C 1 C = 2, x = t C x1 C x, 1 C = 1 C, x = x, x y : = xy + yx = t C xy + t C yx n C x, y1 C, x x = n C x1 C = xx, x + x = t C x1 C, n C x = n C x, t C x = t C x, t C x 2 = t C x 2 2n C x, t C x y = t C xy + t C yx = 2[t C xt C y n C x, y, n C x, ȳ = t C xt C y n C x, y, xy ȳ x = t C x, y n C x, ȳ 1 C.

3 NON-ORTHOGONAL CAYLEY-DICKSON CONSTRUCTION Norm-associative conic algebras. Let C be a conic algebra over k as in 2.1. By [16, Prop , , the following collections of identities are all equivalent in C n C x, yx = t C yn C x, n C x, xy = t C yn C x, n C xy, z = n C x, zy, n C xy, z = n C y, xz, t C xy = n C x, y = t C xt C y n C x, y = t C yx, t C xyz = tc xyz. If they are fulfilled, C is said to be norm-associative. By [16, Prop , normassociative conic algebras are flexible, and their conjugations are algebra involutions Multiplicative conic algebras. Following [16, 19.1, a conic algebra C over k is said to be multiplicative if its norm permits composition: n C xy = n C xn C y for all x, y C. By [16, 19.2 a, multiplicative conic algebras are norm-associative Conic alternative algebras. Let C be a conic alternative k-algebra, so C is a conic algebra satisfying the alternative laws. By [16, , the U-operator of C satisfies the identity 1 U x y = xyx = n C x, ȳx n C xȳ. Note by [16, Exc. 68 that conic alternative algebras will in general not be multiplicative, though they are if the underlying module is projective [16, Prop The classical Cayley-Dickson construction. Let B be any conic algebra over k and µ k an arbitrary scalar. We define a k-algebra C on the direct sum B Bj of two copies of B as a k-module by the multiplication u 1 + v 1 ju 2 + v 2 j := u 1 u 2 + µ v 2 v 1 + v 2 u 1 + v 1 ū 2 j, for u i, v i B, i = 1, 2, and a quadratic form n C : C k by n C u + vj := n B u µn B v u, v B. C together with n C is a conic k-algebra and is said to arise from B, µ by means of the Cayley-Dickson construction, written as CayB, µ in order to indicate dependence on the parameters involved. Note that 1 C = 1 B + 0 j is an identity element for C and that the assignment u u + 0 j gives an embedding, i.e., an injective homomorphism, B C of conic k-algebras, allowing us to identify B C as a conic subalgebra. 3. Elementary properties of the non-orthogonal Cayley-Dickson construction To the best of our knowledge, the first examples of a non-orthogonal Cayley-Dickson construction are due to Pumplün [17 and work over an integral domain whose quotient field has characteristic not 2 by using bases and structure constants closely modeled after the Coxeter octonions [2. Roughly speaking, the non-orthogonal Cayley-Dickson construction proposed in the present paper formalizes the most general way in which a given conic algebra sits in a multiplicative alternative one as a unital subalgebra. The details of this formalization may be read off from the following observation Proposition. The internal construction Let C be a multiplicative conic alternative k-algebra, B a unital subalgebra of C and l C. Then the subalgebra C of C generated by B and l agrees with B + Bl as a k-submodule. More precisely, writing s: B k for the linear form defined by 1 su := n C u, l u B

4 4 HOLGER P. PETERSSON and setting 2 the relations λ := s1 C = t C l k, µ := n C l k, vlu = suv + λvu + vūl, uvl = s vu1 B + suv + s vu λvu + vul, u vl = s vu1 B + s vu + t B uvl, v 1 lv 2 l = λs v 2 v 1 1 B + λsv 1 v 2 + λs v 2 v 1 λ 2 v 2 v 1 + µ v 2 v 1 + s v 2 v 1 1 B sv 1 v 2 + λv 2 v 1 l, n C u + vl = n B u + s vu µn B v hold for all u, u 1, u 2, v, v 1, v 2 B. Proof. Firstly, we simplify notation by writing n := n C resp. t := t C for the norm resp. trace of C. Secondly, we note that the first assertion will follow once we have established the identities 3 7. In order to do so, we begin with 5 and, applying 2.2.5, 2.2.7, 2.3.4, 2.3.5, 1, obtain u vl = tuvl + tvlu nu, vl1 B = tuvl + n v, lu n vu, l1 B = tuvl + s vu s vu1 B, giving 5. Next we use right alternativity linearized to compute vlu+vul = vl u = tlvu+tuvl nu, lv, which implies vlu = tlvu + v [ tu1 B u l nu, lv = suv + λvu + vūl, hence 3. Subtracting 3 from 5 gives 4. Now we proceed to derive 6. Combining 5 with the middle Moufang identity and 2.5.1, we obtain v 1 lv 2 l = v 1 lv 2 l lv 1 v 2 l = sv 1 1 B + λv 1 + tv 1 l v 2 l lv 1 v 2 l = tv 1 lv 2 l + λv 1 v 2 l sv 1 v 2 l lv 1 v 2 l = l v 1 v 2 l + λv 1 v 2 l sv 1 v 2 l = nl, v 2 v 1 l nl v 2 v 1 + λ s v 2 v 1 1 B + sv 1 v 2 + s v 2 v 1 λv 2 v 1 + v 2 v 1 l sv 1 v 2 l = s v 2 v 1 l sv 1 v 2 l + µ v 2 v 1 λs v 2 v 1 1 B + λsv 1 v 2 + λs v 2 v 1 λ 2 v 2 v 1 + λv 2 v 1 l, and this is 6. It remains to establish 7. This follows immediately from multiplicativity, 2.3.4, 1, 2 and the expansion nu+vl = nu+nu, vl+nvnl = nu+s vu µnv The external construction. Let B be a conic k-algebra, s: B k an arbitrary linear form and µ k an arbitrary scalar. Motivated by the formulas derived in Prop. 3.1, we put 1 and write 2 λ := s1 B C := CayB; µ, s := B Bj for the non-associative k-algebra living on the direct sum of two copies of B as a k-module under a bilinear multiplication uniquely determined by the condition that the k-module B, identified in C through the initial summand, is a subalgebra and the equations uvj = s vu1 B + suv + s vu λvu + vuj, vju = suv + λvu + vūj, v 1 jv 2 j = λs v 2 v 1 1 B + λsv 1 v 2 + λs v 2 v 1 λ 2 v 2 v 1 + µ v 2 v 1 + s v 2 v 1 1 B sv 1 v 2 + λv 2 v 1 j

5 NON-ORTHOGONAL CAYLEY-DICKSON CONSTRUCTION 5 hold for all u, v, v 1, v 2 B. One checks immediately that the k-algebra C is unital, with unit element 1 C := 1 B, so the subalgebra B C is, in fact, unital Proposition. Under the assumptions and notation of 3.2, C = CayB; µ, s is a conic k-algebra with unit element, norm, bilinearized norm, trace, conjugation respectively given by C = 1 B, n C u + vj = n B u + s vu µn B v, n C u 1 + v 1 j, u 2 + v 2 j = n B u 1, u 2 + s v 2 u 1 + s v 1 u 2 µn B v 1, v 2, t C u + vj = t B u + s v, u + vj = ū + s v1 B vj for all u, u 1, u 2, v, v 1, v 2 B. We also have 6 for all u, v B. u vj = s vu1 B + s vu + t B uv j Proof. We have seen already in 3.2 that C is unital and 1 holds. Moreover, 2 defines a quadratic form n C on C whose bilinearization is given by 3, and t C := n C 1 C, satisfies 4. Adding to 3.2.4, we arrive at 6. Hence, using 3.2.5, we may compute u + vj 2 = u 2 + u vj + vj 2 = t B uu n B u1 B s vu1 B + s vu + t B uv j λ 2 n B v1 B + λ 2 t B vv λ 2 v 2 + µn B v1 B + λn B v1 B svv + λv 2 j = t B uu + vj + s vu + λt B v sv vj n B u + s vu µn B v 1 B = t B u + s v u + vj n C u + vj1 C = t C u + vju + vj n C u + vj1 C. Thus C is a conic k-algebra with norm, bilinearized norm and trace as indicated. Finally, the formula for the conjugation of C is now obvious Examples. In the situation of 3.2, 3.3, the conic algebra CayB; µ, s over k is said to arise from B, µ, s by the non-orthogonal Cayley-Dickson construction. If s = 0 is the zero linear form, formulas combined with show that the general non-orthogonal Cayley-Dickson construction collapses to the ordinary one: CayB; µ, 0 = CayB, µ. On the other hand, suppose K/k is a purely inseparable field extension of characteristic 2 and exponent at most 1, and suppose further s: K k is a linear form which is unital in the sense that s1 K = 1. Since K as a conic k-algebra has trivial conjugation, a comparison of and with [5, , shows that our general non-orthogonal Cayley-Dickson construction collapses to the one investigated in [5, 4, Proposition. Let C be a multiplicative alternative conic k-algebra, B a conic k- algebra and ϕ: B C a homomorphism of conic algebras. Given an element l C, put µ := n C l k and define a linear form s: B k by 1 su := n C ϕu, l u B. Then there is a unique extension of ϕ to a homomorphism ϕ : CayB; µ, s = B Bj C of conic k-algebras such that ϕ j = l. Proof. Uniqueness follows from the obvious fact that the conic k-algebra C := CayB; µ, s is generated by B and j. To prove existence, we define ϕ : C C by

6 6 HOLGER P. PETERSSON ϕ u + vj = ϕu + ϕvl for all u, v B. Then ϕ is a k-linear map extending ϕ, and we must show and, finally, ϕ uvj = ϕuϕvl, ϕ vju = ϕvlϕu, ϕ v 1 jv 2 j = ϕv 1 lϕv 2 l, n C ϕu + ϕvl = n C u + vj for all u, v, v 1, v 2 B. But setting λ := s1 B = t C l, these formulas follow immediately by comparing and with the relations Corollary. Let B be a conic k-algebra, µ k an arbitrary scalar and s: B k an arbitrary linear form. Put λ := s1 B, C := CayB; µ, s = B Bj as in 3.2 and, given elements a, b B, define a scalar 1 µ := n B a s ba + µn B b k as well as a linear form s : B k by 2 s u := n B u, a + s bu u B. If C is multiplicative alternative, then the linear map ϕ: C := CayB; µ, s = B Bj C defined by 3 ϕu + vj := u + va s bv1 B + svb + s bv λbv + bvj is a homomorphism of conic algebras; moreover, for b B, ϕ is an isomorphism of conic algebras. Proof. ϕ extends the identity of B and satisfies the relation ϕj = l := a + bj. By 3.3.2, 1, 3.3.3, 2 we have n C l = n C a + bj = n B a s ba + µn B b = µ, n C u, l = n C u, a + bj = n B u, a + s bu = s u for all u B. Hence Prop. 3.5 yields a unique homomorphism ψ : C C extending the identity of B and sending j to l. For all u, v B we may apply and 3 to obtain ψu + vj = u + vl = u + va + vbj = u + va s bv1 B + svb + s bv λbv + bvj = ϕu + vj. Hence ϕ = ψ is a homomorphism of conic algebras. Now assume b B. Then 3 shows that ϕ is injective. Its image contains B and l, hence also j = b 1 a + l. But the algebra C is generated by B and j, forcing ϕ to be surjective as well. 4. The conjugation, norm-associativity and flexibility In the first part of this section we generalize [16, Prop by showing that the property of the conjugation of a conic algebra to be an involution is preserved by the non-orthogonal Cayley-Dickson construction. Our approach is based on the following concept A peculiar linear form. Let C be a conic algebra over k. We define a linear form m C : C C k by 1 m C x y := t C xy n C x, ȳ x, y C. By [16, Prop. 18.8, the conjugation of C is an involution if and only if Imm C AnnC.

7 NON-ORTHOGONAL CAYLEY-DICKSON CONSTRUCTION Linear forms on conic algebras. Let B be a conic algebra over k and s: B k any linear form. If we put λ := s1 B, then 2.2.1, 2.2.5, imply for all u, v B. su 2 = t B usu λn B u, suv + svu = t B usv + t B vsu λn B u, v, sū = λt B u su 4.3. Proposition. Let B be a conic k-algebra, µ k and s: B k a linear form. Setting C := CayB; µ, s = B Bj as in 3.2, we have for all u, v, v 1, v 2 B. t C vju = nc vj, ū, t C uvj = nc u, vj, t C v1 jv 2 j = n C v 1 j, v 2 j + µ t B v 2 v 1 n B v 2, v 1 Proof. Beginning with 1, we apply 3.2.4, 3.3.4, 4.2.3, 2.2.6, to obtain t C vju = tc suv + λvu + vūj = sut B v + λt B vu + svū = sut B v + λt B vu + λt B vū svū = sut B v + λt B ut B v t B usv + svu tb = s v1 B v t B u1 B u = s v ū = n C vj, ū, as desired. Next we reduce 2 to 1 by setting x = u, y = vj and applying 2.2.9, 1, , Then t C xy = t C x y t C yx = 2 t C xt C y n C x, y n C y, x = 2n C x, ȳ n C x, ȳ = n C x, ȳ, and we have 2. Finally, turning to 3 and applying 3.2.5, 2.2.3, 3.3.4, 4.2.3, , we obtain t C v1 jv 2 j = t C λs v2 v 1 1 B + λsv 1 v 2 + λs v 2 v 1 λ 2 v 2 v 1 + µ v 2 v 1 + s v 2 v 1 1 B sv 1 v 2 + λv 2 v 1 j = 2λs v 2 v 1 + λsv 1 t B v 2 + λs v 2 t B v 1 λ 2 t B v 2 v 1 + µt B v 2 v 1 + λs v 2 v 1 sv 1 s v 2 + λsv 2 v 1 = λt B v 2 sv 1 + λsv 2 v 1 + λt B v 2 sv 1 + λ 2 t B v 1 t B v 2 λt B v 1 sv 2 λ 2 t B v 2 v 1 + µt B v 2 v 1 λt B v 2 sv 1 + sv 1 sv 2 + λ 2 t B v 2 v 1 λsv 2 v 1 = λt B v 2 sv 1 + λ 2 t B v 1 t B v 2 λt B v 1 sv 2 + µt B v 2 v 1 + sv 1 sv 2. On the other hand, by 3.3.5, 3.3.3, 4.2.3, n C v 1 j, v 2 j = n C v1 j, s v 2 1 B v 2 j = s v 1 s v 2 1 B + µnb v 1, v 2 = s v 1 s v 2 + µn B v 1, v 2 = λt B v 1 sv 1 λt B v 2 sv 2 + µn B v 1, v 2 = λ 2 t B v 1 t B v 2 λt B v 1 sv 2 λt B v 2 sv 1 + sv 1 sv 2 + µn B v 1, v 2,

8 8 HOLGER P. PETERSSON and a comparison with the preceding equation yields Corollary. Imm B = Imm C. Proof. Since B is a direct summand of C as a k-module, B B may be viewed canonically as a submodule of C C, and m B is the restriction of m C to B B. Now, by 4.1 and Prop. 4.3, the ideal Imm C k is generated by the expressions m B u v, m C vj u = 0, mc u vj = 0, for all u, v, v 1, v 2 B. The assertion follows. m C v1 j v 2 j = µm B v2 v Corollary. If the conjugation of B is an involution, then so is the conjugation of C. Proof. Since B and C have the same annihilator, this follows immediately from 4.1 and Cor Next we wish to describe conditions under which the property of a conic algebra to be norm-associative is preserved by the non-orthogonal Cayley-Dickson construction. To this end, we need a conceptual preparation Reminder: associative linear forms. Let A be a non-associative k-algebra. A linear form t: A k is said to be associative if txyz = txyz for all x, y, z A, equivalently, if t vanishes on all associators of A: t[a, A, A = {0}. Clearly, a linear form on an associative algebra is automatically associative. On the other hand, if A is arbitrary but k is a field, non-zero associative linear forms on A exist if and only if [A, A, A A is a proper subspace Alternative linear forms. We continue to consider an arbitrary non-associative k-algebra A. A linear form s: A k is called alternative if the trilinear map A A A k, x, y, z s[x, y, z is alternating, equivalently, if two of the following relations 1 s xxy = sx 2 y, s yxx = syx 2, s xyx = s xyx hold for all x, y A, in which case the third one follows automatically. Clearly, a linear form on an alternative algebra is always alternative. On the other hand, if A is arbitrary but k is a field, non-zero alternative linear forms on A exist if and only if the expressions [x, x, y, [y, x, x for x, y A span a proper subspace of A Proposition. For a norm-associative conic algebra B over k and a linear form s: B k, the following conditions are equivalent. i s is alternative. ii suuv = t B usuv n B usv for all u, v B. iii svuu = t B usvu n B usv for all u, v B. iv suvu = n B u, vsu n B us v for all u, v B. Proof. Since B is flexible by [16, Prop , the third equation of holds. Hence either one of the first two is equivalent to s being alternative. Combined with 2.2.1, this shows that conditions i, ii, iii are equivalent. It remains to establish the implications ii iv iii. ii iv. Setting λ := s1 B and combining ii with 4.2.2, 2.3.5, 2.3.2, we obtain suvu = s uvu = t B uvsu + t B usuv λn B uv, u s uuv and yields iv. = t B uvsu + n B usv λt B vn B u = n B u, vsu + n B u sv λt B v,

9 NON-ORTHOGONAL CAYLEY-DICKSON CONSTRUCTION 9 iv iii. Using iv and 2.2.5, 4.2.1, 4.2.3, we compute s vuu = s u vu suvu = t B usvu + t B vsu 2 n B u, vsu n B u, vsu + n B us v = t B usvu + t B ut B vsu λn B ut B v t B ut B vsu + n B us v = t B usvu n B usv, and this is iii Lemma. Let B be a norm-associative conic k-algebra, s: B k an alternative linear form and λ := s1 B. Then 1 2 s uvw + s wuv = swt B uv + t B wsuv λn B uv, w, s uvw + s uvw = t B usvw t B vswu + t B wsuv + t B vwsu t B wusv + t B uvsw + λ n B uw, v t B vn B u, w for all u, v, w B. Proof. Applying 4.2.2, and Prop. 4.8, we obtain s uvw + s wuv = s uvw + s wuv + s vwu s vwu = s uvw + t B wusv + t B vswu λn B wu, v s vw u + s vuw = t B wusv + t B vswu λn B wu, v t B wsvu t B usvw + n B w, usv + s uvw + vuw = t B wu + n B w, u sv + t B vswu λn B wu, v t B wsvu t B usvw + t B usvw + t B vsuw n B u, vsw = t B wt B usv + t B vsuw + wu λn B wu, v t B wsuv + vu + t B wsuv n B u, vsw = t B wt B usv + t B vt B usw + t B vt B wsu λt B vn B u, w λn B wu, v t B wt B usv t B wt B vsu + λt B wn B u, v + t B wsuv n B u, vsw = t B uvsw + t B wsuv Using 2.3.3, 2.3.4, 2.2.4, we can now compute λ n B wu, v + t B vn B u, w t B wn B u, v. n B wu, v = n B u, wv = n B u v, w = n B u t B v1 B v, t B w1 B w = t B ut B vt B w t B wt B uv t B vn B u, w + n B uv, w = t B wn B u, v t B vn B u, w + n B uv, w.

10 10 HOLGER P. PETERSSON Inserting this into the factor of λ in the final expression of the preceding equation, we end up with 1. Turning to 2 and combining Prop. 4.8 with 4.2.2, we obtain s uvw = s uvw + uwv s uwv = t B vsuw + t B wsuv n B v, wsu s uw v + s vuw = t B vsuw + t B wsuv n B v, wsu t B uwsv t B vsuw + λn B uw, v + s vuw + uvw s uvw = t B wsuv n B v, wsu t B uwsv + λn B uw, v + t B usvw + t B vsuw n B u, vsw s uvw. By again, the sixth summand on the right agrees with t B vsuw = t B vsuw + wu t B vswu = t B ut B vsw + t B vt B wsu λt B vn B w, u t B vswu. Returning with this to the preceding equation, we conclude s uvw = t B wsuv n B v, wsu t B uwsv + λn B uw, v + t B usvw and 2 follows. + t B ut B vsw + t B vt B wsu λt B vn B w, u t B vswu n B u, vsw s uvw = t B usvw t B vswu + t B wsuv + t B vwsu t B wusv + t B uvsw + λ n B uw, v t B vn B u, w s uvw, Theorem. Let B be a conic k-algebra, µ k a scalar and s: B k a linear form. Then the conic algebra CayB; µ, s = B Bj is norm-associative if and only if B is norm-associative and s is alternative. Proof. If C := CayB; µ, s is norm-associative, so is B C as a unital subalgebra. Moreover, given u, v B, we apply 3.3.3, 2.3.4, 3.2.3, and compute s uuv = n C uuv, j = nc v, ūūj = n C v, suū1b + sūū + suū λū 2 + ū 2 j = n C v, λnb ū1 B + λt B ūū λū 2 + ū 2 j = n C v, ū 2 j = n C u 2 v, j = su 2 v = t B usuv n B usv. Thus condition ii of Prop. 4.8 holds, forcing s to be alternative. Conversely, suppose B is norm-associative and s is alternative. Then B is flexible, the conjugation of B is an involution [16, Prop , equations hold in B, and by 2.3 it will be enough to show that holds in C. Actually, by linearity, it suffices to establish the relations 1 2 for all u, v, w B. identities again for all u, v, w B. n C u + vjw, u + vj = nc u + vjt B w, n C u + vjwj, u + vj = nc u + vjt C wj Linearizing we see that 2 is equivalent to the following three n C uwj, u = nb ut C wj, n C uwj, vj + nc vjwj, u = nc u, vjt C wj, n C vjwj, vj = nc vjt C wj,

11 NON-ORTHOGONAL CAYLEY-DICKSON CONSTRUCTION 11 We begin with the verification of 1. Applying 3.2.4, 3.3.2, 3.3.3, we obtain n C u+vjw, u + vj nc u + vjt B w [ = n C [uw swv + λvw + v wj, u + vj nb u + s vu µn B v t B w = n B uw, u swn B v, u + λn B vw, u + s vuw sws vv + λs vvw + s w vu µn B v w, v n B ut B w s vut B w + µn B vt B w Here norm-associativity of B by implies n B uw, u = n B ut B w, n B v w, v = n B vt B w = n B vt B w. On the other hand, we trivially have sws vv = λn B vsw, while Prop. 4.8 yields s vvw = t B vsvw svvw = n B vsw since s is alternative. Finally, we deduce from 4.9.1, s vuw + s w vu = swt B vu + t B ws vu λn B vu, w = swn B v, u + t B ws vu λn B u, vw. Inserting all this into the final expression of the displayed chain of equations above, it follows that this expression is zero, hence that 1 holds. We now proceed to deduce 3 by using 3.2.3, 3.3.4, and Prop. 4.8 iv combined with 2.3.2, which imply n C uwj, u nb ut C wj [ = n C s wu1b + suw + s wu λwu + wuj, u n B us w = s wut B u + sun B w, u + 2s wn B u λn B wu, u + s ū wu n B us w = s wut B u + sun B w, u + s wn B u λn B wu, u + t B us wu n B u, wsu + n B usw = λt B wn B u λn B wu, u = 0, and the proof of 3 is complete. We now come to the verification of 4, which is the most involved one of all. We begin by using 3.2.3, 3.2.5, to manipulate the left-hand side as follows: n C uwj, vj + nc vjwj, u [ = n C s wu1b + suw + s wu λwu + wuj, vj [ + n C λs wv1b + λsvw + λs wv λ 2 wv + µ wv + [ s wv1 B svw + λwv j, u = s wus v + sus vw + s ws vu λs vwu µn B wu, v λs wvt B u + λsvn B w, u + λs wn B v, u λ 2 n B wv, u + µn B wv, u + s wvsu svs wu + λs v wu. Here µn B wv, u = µn B v, wu cancels against the fifth summand on the very right of the preceding expression. Moreover, s wus v svs wu = λs wut B v, sus vw + s wvsu = λsut B vw = λsun B v, w, while s ws vu = n C u, vjt C wj. Hence n C uwj, vj + nc vjwj, u = nc u, vjt C wj + λα,

12 12 HOLGER P. PETERSSON where α := s wut B v + sun B v, w s wvt B u + svn B w, u + s wn B v, u λn B wv, u + s vut B w s vwu + s vwu It suffices to show α = 0. obtain To this end, we apply 4.9.2, 2.2.4, 2.3.2, and s vwu + s vwu = t B vswu t B wsu v + t B us vw Returning to the definition of α, we conclude + t B wus v t B u vsw + t B vwsu + λ n B vu, w t B wn B v, u = swut B v su vt B w + s vwt B u + s vt B wu swn B u, v + sun B v, w + λ n B u, vw t B wn B v, u. α = s wut B v + sun B v, w s wvt B u + svn B w, u + s wn B v, u λn B wv, u + s vut B w swut B v + su vt B w s vwt B u s vt B wu + swn B u, v sun B v, w λn B u, vw + λt B wn B v, u = s wut B v + swut B v + λt B vt B wu + sv t B wu + n B w, u + sun B v, w sun B v, w s wv + s vw t B u + s w + sw n B u, v λn B u, vw + wv + s vu + su v + λn B u, v t B w = sut B vt B w λt B vt B wu + svt B wt B u λt B un B v, w + λt B wn B u, v λt B vn B w, u λt B wn B u, v + λt B un B v, w + sut B vt B w + s vt B wt B u λn B u, vt B w + λn B u, vt B w = λt B v t B wu + n B w, u + sv + s v t B wt B u = λt B vt B wt B u + λt B vt B wt B u = 0, and the proof of 4 is complete. It remains to deal with 5: by 3.2.5, 3.3.3, we have [ n C vjwj, vj = nc λs wv1b + λsvw + λs wv λ 2 wv + µ wv + [ s wv1 B svw + λwv j, vj = λs wvs v + λsvs vw + λs ws vv λ 2 s vwv + µs v wv µs wvt B v + µsvn B w, v λµn B wv, v Here λs wvs v = λ 2 s wvt B v λs wvsv, λs ws vv = λ 2 s wn B v. Moreover, since s is alternative, Prop. 4.8 implies s vwv = t B vswv svwv = swvt B v n B v, wsv + n B vs w and, similarly, s v wv = s wvt B v n B v, wsv + n B vsw.

13 NON-ORTHOGONAL CAYLEY-DICKSON CONSTRUCTION 13 Therefore n C vjwj, vj = λ 2 s wvt B v + λs wvsv + λs vwsv + λ 2 s wn B v λ 2 swvt B v + λ 2 n B v, wsv λ 2 n B vs w + µs wvt B v µn B v, wsv + µn B vsw µs wvt B v + µsvn B v, w λµn B wv, v = λ 2 svt B vt B w + λ 2 svn B v, w + λ 2 svn B v, w+ + µ sw λt B w n B v = µs wn B v = n C vjt C wj. Thus 5 holds and the theorem is proved Killing the annihilator. Let B be a conic k-algebra and a := AnnB. We put k := k/a, write π : k k, α α, for the natural projection and, following [14, 2.9, have the canonical identification as conic k -algebras such that B := B k = B/aB = B u α = αu, u = u 1 k α k, u B. More specifically, the k -module structure of B is given by α u = αu α k, u B and norm, bilinearized norm, trace of B act on B according to the rules n B u = n B u, n B u, v = n B u, v, t B u = t B u for all u, v B. In particular, B = B as Z-algebras and ι B = ι B as Z-linear maps since, for example, ι B u = t B u1 B u = t B u 1 B u = t B u1 B u = ι B u for all u B. Now fix a scalar µ k and a linear form s: B k. Then s := π s: B = B k is the scalar extension of s from k to k, and since the non-orthogonal Cayley-Dickson construction obviously is compatible with base change, we conclude CayB; µ, s = CayB; µ, s = CayB ; µ, s = CayB; µ, s as conic k -algebras. With an eye on Thm we note that s is alternative if and only if, for u, v, w B, the expression s[u, v, w belongs to AnnB as soon as two of the three arguments u, v, w coincide Theorem. Let B be a conic k-algebra, µ k and s: B k a linear form. With the notation and conventions of 4.11, the following conditions are equivalent. i CayB; µ, s is flexible. ii B is flexible, the conjugation of B is an involution and s is alternative. Proof. Put C := CayB; µ, s i ii. Since C is flexible, so is B and tc xy n C x, ȳ x = n C x, xy t C yn C x 1 C

14 14 HOLGER P. PETERSSON for all x, y C [16, Prop b. In particular, for u 1, u 2, v 1 B and x := u 1 + v 1 j, y := u 2, we may apply to conclude that tc xy n C x, ȳ x = t B u 1 u 2 n B u 1, ū 2 + t C v1 ju 2 nb v 1 j, ū 2 u 1 + v 1 j = t B u 1 u 2 n B u 1, ū 2 u 1 + v 1 j belongs to k1 B. Comparing Bj-components, we deduce t B u 1 u 2 n B u 1, ū 2 AnnB. Hence the conjugation of B is an involution [16, Prop a, and it remains to show that s is alternative. Combining Cor. 4.5 with what has been said in 4.11, we first note that C is flexible and its conjugation is an involution, forcing C to be norm associative by [16, Cor since AnnC = AnnB = {0}. Now Thm shows that s is alternative. ii i. By hypothesis and 4.11, the k -algebra B is flexible and its conjugation is an involution. Since, therefore, B is norm-associative [16, Cor , so is C by Thm since s is alternative by hypothesis. Thus C is flexible. 5. Commutativity In this section, we wish to find out under what circumstances the non-orthogonal Cayley-Dickson construction leads to commutative algebras. In the classical orthogonal case, this is known to happen if and only if the conic algebra entering into the construction is commutative itself and has trivial conjugation [16, Thm a. Extending this to the general non-orthogonal case turns out to be easy Commutator relations. Let B be a conic k-algebra, µ k a scalar, s: B k a linear form and C := CayB; µ, s = B Bj the corresponding non-orthogonal Cayley- Dickson construction as in 3.2. With λ := s1 B, we claim 1 2 [u, vj = s vu1 B + 2suv + s vu 2λvu + vu ū j, [v 1 j, v 2 j = λs v 1 v 2 v 2 v 1 1 B + λsv 1 v 1 v 2 λsv 2 v 2 v 1 + λ 2 [v 1, v 2 + µ v 2 v 1 v 1 v 2 + s v 2 v 1 v 1 v 2 1 B + sv 2 v 1 sv 1 v 2 λ[v 1, v 2 j for all u, v, v 1, v 2 B. Indeed, a straightforward verification based on 3.2.4, implies [u, vj = uvj vju = s vu1 B + suv + s vu λvu + vuj + suv λvu vūj = s vu1 B + 2suv + s vu 2λvu + vu ū j, hence 1, while yields hence 2. [v 1 j, v 2 j = v 1 jv 2 j v 2 jv 1 j = λs v 2 v 1 1 B + λsv 1 v 2 + λs v 2 v 1 λ 2 v 2 v 1 + µ v 2 v 1 + s v 2 v 1 1 B sv 1 v 2 + λv 2 v 1 j + λs v1 v 2 1 B λsv 2 v 1 λs v 1 v 2 + λ 2 v 1 v 2 µ v 1 v 2 s v 1 v 2 1 B sv 2 v 1 + λv 1 v 2 j = λs v 1 v 2 v 2 v 1 1 B + λsv 1 v 1 v 2 λsv 2 v 2 v 1 + λ 2 [v 1, v 2 + µ v 2 v 1 v 1 v 2 + s v 2 v 1 v 1 v 2 1 B + sv 2 v 1 sv 1 v 2 λ[v 1, v 2 j,

15 NON-ORTHOGONAL CAYLEY-DICKSON CONSTRUCTION Lemma. With the notation and assumptions of 5.1, assume that B is commutative. Then the following conditions are equivalent. i suv = svu for all u, v B. ii su1 B = λu for all u B. iii suv = λuv for all u, v B. Proof. i ii. Setting v = 1 B in i gives ii. ii iii. If ii holds, then suv = su1 B v = λuv for all u, v B, giving iii. iii i. Since the right-hand side of iii is symmetric in u, v, so is the left, whence i holds Reduction modulo one. Let B be a conic algebra over k. Following Loos [10, 1.2 with a slightly different notation, we put Ḃ := B/k1 B as a k-module and denote by u u the natural map from B to Ḃ. Note that, if B is a finitely generated projective k-module of rank n + 1, then Ḃ is is one of rank n since 1 B B is unimodular [16, Proposition. Let B be a faithful and commutative conic k-algebra. Then the assignment s λ := s1 B defines a linear isomorphism from the k-module of linear forms on B satisfying the equivalent conditions i iii of Lemma 5.2 onto the annihilator of Ḃ. Proof. If s: B k is a linear form satisfying the equivalent conditions of Lemma 5.2 and λ := s1 B, then condition ii of the lemma implies λ u = su 1 B = 0, so λ annihilates Ḃ. The assignment s λ clearly defines a linear map, which by ii and faithfulness is injective. Conversely, suppose λ k annihilates Ḃ. Then λb k1 B, and by faithfulness again, there is a unique map s: B k satisfying condition ii of Lemma 5.2. Clearly, s is linear and s1 B = λ Proposition. Let B be a conic k-algebra, µ k a scalar and s: B k a linear form. Then the following conditions are equivalent. i The non-orthogonal Cayley-Dickson construction CayB; µ, s is commutative. ii B is commutative with trivial conjugation and suv = svu for all u, v B. Proof. We put C := CayB; µ, s = B Bj as in 3.2. ii i. If ii holds, then an inspection of shows [v 1 j, v 2 j = 0 for all v 1, v 2 B. Moreover, given u, v B, not only vu ū = 0 but also, by Lemma 5.2, s vu1 B + 2suv + s vu 2λvu = λuv + 2λuv + λuv 2λuv = 0, hence [u, vj = 0 by Hence C is commutative. i ii. Since C is commutative, so is B as unital subalgebra. Moreover, for v = 1 B shows u = ū for all u B, so the conjugation ι B is the identity. Combining all this with 5.1.2, we conclude 0 = [v 1 j, v 2 j = sv 2 v 1 sv 1 v 2 j for all v 1, v 2 B, and also the final assertion of ii follows. 6. Associativity In this section, we wish to find necessary and sufficient conditions for the output of a non-orthogonal Cayley-Dickson construction to be an associative algebra. In the classical orthogonal case, this is known to happen if and only if the conic algebra entering into the construction is commutative associative and its conjugation is an involution [16, Thm b. In the general case we will see that a simple additional property of the linear form involved will be enough to guarantee the same conclusion.

16 16 HOLGER P. PETERSSON 6.1. Nuclei. Let A be a non-associative k-algebra. Beside the ordinary nucleus [16, it is sometimes useful to consider one-sided nuclei that are respectively defined by 1 Nuc l A := { x A [x, A, A = {0} } left nucleus, 2 Nuc m A := { x A [A, x, A = {0} } middle nucleus, 3 Nuc r A := { x A [A, A, x = {0} } right nucleus. They are obviously k-submodules of A. But using the associator identity 4 [xy, z, w [x, yz, w + [x, y, zw = x[y, z, w + [x, y, zw, valid in arbitrary non-associative algebras [16, 8.5.2, it follows immediately that they are, in fact, subalgebras of A. There are slight modifications of the preceding concepts depending on the choice of a subalgebra B A. We define Nuc l B A := { x A [x, B, B = {0} }, Nuc m B A := { x A [B, x, B = {0} }, Nuc r B A := { x A [B, B, x = {0} }, which are submodules, but, in general, no longer subalgebras, of A. On the other hand, we have 6.2. Lemma. Let A be a non-associative k-algebra and B A an associative subalgebra. Then Nuc l B AB Nuc l B A. Proof. Let x Nuc l B A and y, z, w B. In the associator identity 6.1.4, the second resp. third summand on the left vanishes because B A is a subalgebra and so yz resp. zw belongs to B. On the other hand, the first summand on the right of vanishes because B is an associative algebra, while the second summand does because x belongs to Nuc l B A. Thus [xy, z, w = 0, forcing xy Nuc l B A Returning to the non-orthogonal Cayley-Dickson construction. We now return to our conic k-algebra B, a scalar µ k and a linear form s: B k to form the non-orthogonal Cayley-Dickson construction C := CayB; µ, s = B Bj as in 3.2; in particular, we put λ := s1 B. While the algebra C will in general not be alternative, there are certain nice elements that behave as if it were Lemma. With the assumptions and notation of 6.3, the relations hold for all u, v B. ju = su1 B + λu + ūj, jvj = µ v + s vj, vjj = µv + λvj Proof. All three equations follow from by a straightforward computation: as claimed. ju = 1 B ju = su1 B + λu + ūj, jvj = 1 B jvj = λs v1 B + λ 2 v + λs v1 B λ 2 v + µ v + s v1 B λv + λv j = µ v + s v1 B j, vjj = vj1 B j = λsv1 B + λsv1 B + λ 2 v λ 2 v + µv + sv1 B sv1 B + λv j = µv + λvj,

17 NON-ORTHOGONAL CAYLEY-DICKSON CONSTRUCTION Remark. By we have n B j = µ, n B j, v = s v, t B j = λ. Hence resp amounts to jvj = n B j, vj n B j v resp. vjj = n B jv +t B jvj = vt B jj n B j1 B = vj 2, in agreement with 2.5.1, the formula for the U-operator in conic alternative algebras resp. the right alternative law Theorem. With the assumptions and notation of 6.3, the following conditions are equivalent. i The conic algebra C = CayB; µ, s is associative. ii The conic algebra B is commutative associative, its conjugation is an involution and the linear form s satisfies the relation 1 suv1 B = suv + svū λūv for all u, v B. Proof. We may assume that B is associative and, by Thm. 4.12, that its conjugation is an involution, which by Thm again implies that C is flexible since s is trivially alternative, so we have 2 [x, y, x = 0, [x, y, z = [z, y, x for all x, y, z C In order to prove the theorem, we will investigate the conditions that are necessary and sufficient for the associators [a, b, c to vanish identically in a, b, c C. By trilinearity, we may assume a, b, c B Bj and thus are left with the following cases. Case 1. a, b, c B. Then [a, b, c = 0 since we have assumed that B is associative. Case 2. {a, b, c} Bj = 1. Case 2.1. a Bj, b, c B. Then a = vj, b = u 1, c = u 2 for some u 1, u 2, v B. Since vj j v mod B by 6.4.1, we have [a, b, c = [vj, u 1, u 2 = [j v, u 1, u 2 in view of Case 1, so we have to find necessary and sufficient conditions for j v to belong to Nuc l B A. By Lemma 6.2, we may assume v = 1 B. Using 6.4.1, 3.2.4, we now compute [a, b, c = [j, u 1, u 2 = ju 1 u 2 ju 1 u 2, ju 1 u 2 = [ su 1 1 B + λu 1 + ū1 j u 2 = [ su 1 u 2 + λu 1 u 2 + ū1 ju 2 = [ su 1 u 2 + λu 1 u 2 su 2 ū 1 + λū 1 u 2 + [ū1 ū 2 j, ju 1 u 2 = [ su 1 u 2 1 B + λu 1 u 2 + u1 u 2 j = [ su 1 u 2 1 B + λu 1 u 2 + [ū2 ū 1 j. Comparing we see that [a, b, c = 0 for all possible choices of Case 2.1 if and only if B is commutative and su 1 u 2 1 B = su 1 u 2 +su 2 ū 1 λū 1 u 2 for all u 1, u 2 B, equivalently, B is commutative associative and 1 holds. In particular, we have established the implication i ii of the theorem, and it remains to establish the implication ii i. For the remainder of the proof, we therefore assume that B is commutative associative and 1 holds. We must show that C is associative, i.e., [a, b, c = 0 for all a, b, c B Bj. By flexibility 2, the discussion of Case 2 will be complete once we have dealt with Case 2.2. b Bj, a, c B. Then a = u 1, b = vj, c = u 2 for some u 1, u 2, v B. We now combine the associator identity with and 2 to conclude that [a, b, c = [u 1, vj, u 2 = [x, yz, w with x = u 1, y = v, z = j, w = u 2 is a Z-linear

18 18 HOLGER P. PETERSSON combination of [xy, z, w = [u 1 v, j, u 2 Case 2.2 with v = 1 B, [x, y, zw = [u 1, v, ju 2 = [ū 2 j, v, u 1 Case 2.1, x[y, z, w = u 1 [v, j, u 2 Case 2.2 with v = 1 B, [x, y, zw = [u 1, v, ju 2 = [j, v, u 1 u 2 Case 1. Hence we may assume v = 1 B. After this reduction we use 3.2.4, to compute [a, b, c = [u 1, j, u 2 = u 1 ju 2 u 1 ju 2, u 1 ju 2 = [ su 2 u 1 + λu 1 u 2 + [u1 ū 2 j, u 1 ju 2 = u 1 [ su2 1 B + λu 2 + ū2 j = [ su 2 u 1 + λu 1 u 2 + u1 ū 2 j = [ su 2 u 1 + λu 1 u 2 + [ su2 u 1 1 B + su 1 ū 2 + su 2 u 1 λū 2 u 1 + [ū2 u 1 j = [ su 2 u 1 1 B + su 1 ū 2 λū 2 u 1 + λu 1 u 2 + [u1 ū 2 j. Comparing the final expressions of the last two equations by means of 1, we see that they are the same, forcing [a, b, c = 0, as desired. Case 3. {a, b, c} Bj = 2. Case 3.1. a, b Bj, c B. Then a = v 1 j, b = v 2 j, c = u for some u, v 1, v 2 B. This time combining the associator identity with 2, we conclude that [a, b, c = [v 1 j, v 2 j, u = [xy, z, w with x = v 1, y = j, z = v 2 j, w = u is a Z-linear combination of x[y, z, w = v 1 [j, v 2 j, u Case 3.1 with v 1 = 1 B, [x, y, zw = [v 1, j, v 2 ju = [v 2 j, j, v 1 u Case 3.1 with v 2 = 1 B, [x, yz, w = [v 1, jv 2 j, u = s v 2 [v 1, j, u Case 2.2, [x, y, zw = [v 1, j, zw = [zw, j, v 1 Case 2.2, and Case 3.1 with v 2 = 1 B. Hence we may assume v 1 = 1 B or v 2 = 1 B. Case v 1 = 1 B. Then Using 6.4.2, 6.4.1, we compute a = j, b = vj, c = u for some u, v B. [a, b, c = [j, vj, u = jvju j vju, jvju = µ v + s vj u = µ vu + s vju = µ vu + s v [ su1 B + λu + ūj = [ sus v1 B + λs vu + µu v + [ s vū j, j vju = j [ suv + λvu + [vūj = sujv + λjvu + jvūj = [ susv1 B λsuv [ su v j + [ λsvu1 B + λ 2 vu + [λū vj + µu v + su vj = [ susv λsuv 1 B λsuv + λ 2 uv + µu v + [ su v1 B su v + λū v j.

19 NON-ORTHOGONAL CAYLEY-DICKSON CONSTRUCTION 19 Comparing by means of 1, we obtain [a, b, c = [ λsuv1 B susv1 B sus v1 B + λsuv + λs vu λ 2 uv as desired. Case v 2 = 1 B. Then [ su v1 B su v s vū + λū v j = λsuv1 B λt B vsu1 B + λsuv + λs vu λ 2 uv = λsuv1 B λsu v + λ 2 t B vu λsvu λ 2 uv = λ svu1 B svu su v + λ vu = 0, Using 6.4.3, 3.2.4, 6.4.1, we compute a = vj, b = j, c = u for some u, v B. [a, b, c = vjj u vjju, vjj u = µv + λvj u = µvu + λvju = µuv + [ λsuv + λ 2 vu + λvūj = [ λsuv + λ 2 + µuv + [λūvj, vjju = vj [ su1 B + λu + ūj = suvj + λvju + vjūj Comparing and using 1, we conclude = [ suv j + [ λsuv + λ 2 vu + [λvūj + [ λsuv1 B + λsvū + λsuv λ 2 ūv + µuv + [ suv1 B svū + λūv j = [ λsuv1 B + λsvū + λ 2 u ūv + µuv + [ suv1 B suv svū + 2λūv j [a, b, c = [ λsuv1 B λsuv λsvū + λ 2 uv λ 2 u ūv as desired. Case 3.2. a, c Bj, b B. Then + [ λūv suv1 B + suv + svū 2λūv j = λ [ suv1 B suv svū + λūv [ suv1 B suv svū + λūv j = 0, a = v 1 j, b = u, c = v 2 j for some u, v 1, v 2 B. Combining the associator identity with 2, we conclude that [a, b, c = [v 1 j, u, v 2 j = [xy, z, w with x = v 1, y = j, z = u, w = v 2 j is a Z-linear combination of x[y, z, w = v 1 [j, u, v 2 j = v 1 [v 2 j, u, j Case 3.2 with v 2 = 1 B, [x, y, zw = [v 1, j, uw Case 2.2, [x, yz, w = [v 1, ju, v 2 j Cases 2.1, 3.1 because of and 2, [x, y, zw = [v 1, j, zw Cases 2.2, 3.1 because of 2. We may thus assume v 2 = 1 B. Then a = vj, b = u, c = j for some u, v B.

20 20 HOLGER P. PETERSSON Using 3.2.4, 6.4.3, we compute [a, b, c = [vj, u, j = vju j vjuj, vju j = [ suv + λvu + [vūj j = [ suv + λuv j + [vūj j = [ suv + λuv j + µūv + [λūvj = µūv + [ suv + λt B uv j = [µūv + [sūvj, vjuj = [ λsūv1 B + λsvu + λsūv λ 2 uv + µūv [ sūv1b svu + λuv j, which by 1 implies [a, b, c = λ [ sūv1 B sūv svu + λuv [ sūv1 B sūv svu + λuv j = 0, as desired. Case 4. a, b, c Bj. Then a = v 1 j, b = v 2 j, c = v 3 j for some v 1, v 2, v 3 B. Again we make use of the associator identity combined with 2, to conclude that [a, b, c = [v 1 j, v 2 j, v 3 j = [x, yz, w with x = v 1 j, y = v 2, z = j, w = v 3 j is a Z-linear combination of x[y, z, w = v 1 j[v 2, j, v 3 j = v 1 j[v 3 j, j, v 2 Case 3.1, [x, y, zw = [v 1 j, v 2, jw Case 3.2, [xy, z, w = [v 1 jv 2, j, v 3 j = [v 3 j, j, v 1 jv 2 Case 3.1, Case 4 for v 2 = 1 B, [x, y, zw = [v 1 j, v 2, jv 3 j Cases 2.1, 3.2. We are thus reduced to the case v 2 = 1 B and then have a = vj, b = j, c = wj for some v, w B. Using 6.4.3, 3.2.3, 6.4.2, 3.2.4, we compute [a, b, c = [vj, j, wj = vjj wj vjjwj, vjj wj = µv + [λvj wj = µvwj + λvjwj = [ µs wv1 B + µsvw + µs wv λµwv + [µwvj + [ λ 2 s wv1 B + λ 2 svw + λ 2 s wv λ 3 wv + λµ wv + [ λs wv1 B λsvw + λ 2 wv j = [ λ 2 + µ s wv1 B s wv svw + λwv + λµv w + [ µvw + λ s wv1 B svw + λwv j = [λµv w + [ λs wv + µvw j, vjjwj = vj µ w + s wj = µvj w + s wvjj = [ µs wv + λµv w + [µvwj + [ µs wv + [ λs wv j = [λµv w + [ λs wv + µvw j Comparing, we conclude [a, b, c = 0, which completes the proof of the theorem Reminder: quadratic algebras. Following Knus [7, 1.3.6, a k-algebra R is said to be quadratic if it contains a unit element and is finitely generated projective of rank 2 as a k-module. In this case, R is a conic algebra, with norm, trace respectively given by n R x = detl x, t R x = trl x for all x R.

21 NON-ORTHOGONAL CAYLEY-DICKSON CONSTRUCTION Corollary. Let R be a quadratic k-algebra, µ k a scalar and s: R k a linear form. Then the conic algebra CayR; µ, s is associative. Proof. Since R is commutative associative and its conjugation is an involution,, it will be enough by Thm. 6.6 to show that s satisfies equation Localizing if necessary, we may assume that R is a free k-module of rank 2, with basis 1 C, w, for some w R. Thanks to bilinearity, it suffices to establish for u = 1 C, v = 1 C and u = v = w. Indeed, s1 R v + sv 1 R λ 1 R v = λv + sv1 R λv = s1 R v1 R, su1 R + s1 R ū λū1 R = su1 R + λū λū = su1 R 1 R, as claimed. sww + sw w λ ww = t R wsw1 R λn R w1 R = s t R ww n R w1 R 1R = sw 2 1 R, 6.9. Corollary. Let B be a commutative associative conic k-algebra whose conjugation is an involution and µ k a scalar. Then the conic algebra CayB; µ, t B is associative. Proof. By Thm. 6.6, it suffices to show that s := t B satisfies equation Since λ = s1 B = t B 1 B = 2, we have t B uv + t B vū 2ūv = t B uv + t B vū ū v = t B uv + t B vū t B ūv t B vū + n B ū, v1 B as claimed. = t B uv + t B vū t B uv t B vū + t B uv1 B = t B uv1 B, Corollary. Let B be a commutative associative conic k-algebra that is projective of rank at least 3. Suppose further µ k is a scalar and s: B k is a linear form such that the conic algebra CayB; µ, s is associative. Then sut B v = svt B u for all u, v B. In particular, if in addition to the above t B is surjective, then s = αt B for some α k. Proof. By Thm. 6.6, the linear form s satisfies the equation 1 suv1 B = suv + svū λūv λ := s1 B for all u, v B. Since the left-hand side is symmetric in u, v, so is the right and we have suv + svū λūv = svu + su v λ vu. This also follows from applying the conjugation to 1. We now conclude suv v svu ū λūv u v = 0. But since ūv u v = t B uv uv t B vu+uv = t B uv t B vu, u ū = u t B u1 B +u = 2u t B u1 B and, similarly, v v = 2v t B v1 B, we obtain su 2v t B v1 B sv 2u tb u1 B λ tb uv t B vu = 0. Writing the left-hand side as a linear combination of v, u, 1 B, we finally end up with 2su λtb u v 2sv λt B v u sut B v svt B u 1 B = 0. 2 Since the linear form u 2su λt B u obviously kills 1 B, the expression 2s u λt B u makes sense for u B although the individual terms 2s u and λt B u do not. Thus, reading 2 in Ḃ, we conclude 3 2s u λtb u v = 2s v λt B v u u, v Ḃ.

22 22 HOLGER P. PETERSSON Localizing if necessary, we may assume that Ḃ is free of rank at least 2 as a k-module. Picking a basis ė i i I, I 2, of Ḃ, equation 3 implies 2sė i λt B e i = 0 = 2sė j λt B ė j for all i, j I, i j. Hence 2su = λt B u for all u B, and since 1 B is unimodular [16, 18.6, 2 yields the first assertion of the corollary: sut B v = svt B u for all u, v B. If t B is surjective, some v B has t B v = 1, and the second assertion follows as well. 7. Alternativity This section is devoted to the problem of finding conditions that are necessary and sufficient for the output of the non-orthogonal Cayley-Dickson construction to be an alternative conic algebra. In the classical orthogonal case, this is known to happen if and only if the conic algebra entering into the construction is associative and its conjugation is an involution [16, Thm c. In the general case, a certain alternating trilinear map will have to vanish identically in order to reach the same conclusion. Throughout this section, we fix a conic k-algebra B, a scalar µ k and a linear form s: B k in order to from the conic algebra C := CayB; µ, s = B Bj as in 3.2. As usual, we put λ := s1 B Proposition. If B is multiplicative alternative, then H B,s : B 3 B defined by 1 H B,s u, v, w = s uvw 1 B + suv w + svwū su wv + suvw svū w swūv + λūv w for all u, v, w B is an alternating trilinear map such that 2 for all u, v B. H B,s u, v, 1 B = H B,s u, v, uv = 0 Proof. The map H B,s is clearly trilinear, so it remains to show that it is alternating and satisfies 2. Noting that the conjugation of B by [16, Prop is an involution, we begin with the former by abbreviating H := H B,s, using and computing Hu, u, w = t B usuw1 B + n B usw1 B + t B usu w λn B u w + suwū By 2.2.4, we have Hence Similarly, su wu + suuw suū w n B usw1 B + λn B u w. t B usuw1 B + suwū = suwu, t B usu w suū w = suu w. Hu, u, w = suwu + suu w su wu + suuw = t B wsuu + t B wsuu = 0. Hu, v, v = t B vsuv1 B + n B vsu1 B + suv v + t B vsvū λn B vū su vv + t B vsuv n B vsu1 B svū v svūv + λn B vū. Since t B vsuv1 B + suv v = suvv, su vv + t B vsuv = suvv, svū v svūv = t B vsvū, we have Hu, v, v = suvv + suvv = 0, and summing up it follows that H is alternating. We now verify 2. First of all, Hu, v, 1 B = suv1 B + suv1 B + svū suv + suv svū λūv + λūv = 0. Moreover, Hu, v, uv = t B uvsuv1 B + λn B uv1 B + suvuv + svuvū su vūv + suvuv svū vū suvūv + λūv vū.

23 NON-ORTHOGONAL CAYLEY-DICKSON CONSTRUCTION 23 Here we combine the relation t B uvsuv1 B + suvuv = suvuv with to obtain Hu, v, uv = suvuv + λn B uv1 B + n B v, ūsvū n B vsūū t B usu vv + n B u, vsuv n B usvv + n B v, ūsuv n B vsuū n B u, vsvū + n B usvv suvūv + λn B vt B uū λn B vn B u1 B. Canceling out the second resp. third, seventh term against the last resp. tenth, eleventh one on the right-hand side and regrouping, we obtain Hu, v, uv = suvuv t B usu vv suvūv n B vsūū n B vsuū + λn B vt B uū + n B u, vsuv + n B ū, vsuv = t B ut B vsuv λn B vt B uū + λn B vt B uū + t B ut B vsuv, and this is zero as claimed Local generation. Let r be a natural number. A unital non-associative algebra A over k is said to be locally generated by r elements if, for every p Speck, the k p -algebra A p is unitally generated by r elements in the usual sense. For example, a quadratic k-algebra cf. 6.7 is locally generated by a single element. Now suppose B is a quaternion algebra over k [16, Locally, B arises from a quadratic étale algebra by means of the Cayley-Dickson construction [16, Cor Hence quaternion algebras are locally generated by two elements Corollary. If B as in 7.1 is locally generated by two elements, then H B,s = 0. Proof. The assertion is local on k, so we may assume that k is a local ring. If B is unitally generated by x, y B, then [16, Exc. 70 shows that it is spanned by 1 B, x, y, xy as a k-module. Since H := H B,s is alternating trilinear, the assertion will follow once we have shown Hu, v, w = 0 for distinct elements u, v, w {1 B, x, y, xy}. But in view of Prop. 7.1, this is obvious Theorem. If B is associative and its conjugation is an involution, then 1 [u 1 + v 1 j, u 1 + v 1 j, u 2 + v 2 j = H B,s u 1, v 1, u 2 + H B,s ū 1, v 1, v 2 j for all u 1, u 2, v 1, v 2 B. Proof. We proceed in several steps Since B is associative, every linear form on B as well B cf is trivially alternative cf. 4.7, and as B is flexible, so is C, by Thm With H := H B,s we now claim that it suffices to show [u, u, w = 0, [vj, u, u = 0, [vj, vj, u = 0, [wj, vj, vj = 0, [u, vj, w + [vj, u, w = Hu, v, w, [wj, u, vj + [wj, vj, u = Hū, v, wj