Journal of 243, 41᎐68Ž. 2001 doi:10.1006rjabr.2001.8773, available online at http:rrwww.idealibrary.com on

On the Tensor Product of Composition

Patrick J. Morandi

Department of Mathematical Sciences, New Mexico State Uni¨ersity, Las Cruces, New Mexico 88003 E-mail: [email protected]

Jose´´ M. Perez-Izquierdo1

Departamento de Matematicas´´ y Computacion, Uni¨ersidad de La Rioja, 26004 Logrono,˜ Spain E-mail: [email protected]

and

S. Pumplun¨

Facultat fur¨¨ Mathematik, Uni¨ersitat Regensburg, Uni¨ersitatstrasse 31, D-93040 Regensburg, Germany E-mail: [email protected]

Communicated by Georgia Benkart

Received December 10, 1999

DEDICATED TO H. P. PETERSSON ON THE OCCASION OF HIS 60TH BIRTHDAY

1. INTRODUCTION

m Let C1 F C2 be the tensor product of two composition algebras over a F with charŽ.F / 2. Brauerwx 8 and Albert w 1᎐3 x seemed to be the first mathematicians who investigated the tensor product of two quater- nion algebras. Later their results were generalized to this more general

1 The second author thanks DGES for from grants PB-1291-C03-02 and API- 00rB04.

41 0021-8693r01 $35.00 Copyright ᮊ 2001 by Academic Press All rights of reproduction in any form reserved. 42 MORANDI, PEREZ´¨-IZQUIERDO, AND PUMPLUN situation by Allisonwx 4᎐6 and to algebras over rings by Knus wx20 . In Section 2 we give some new results on the Albert form of these algebras. We also investigate the F-quadric defined by this Albert form, generalizing a result of Knuswx 21 . ␴ s ␥ m ␥ Since Allison regarded the 12as an essential part s m of the algebra C C1 F C2 , he only studied of C which are compatible with ␴ . In Section 3 we determine, if charŽ.F / 2, the of a tensor product of algebras. We also show that any automorphism of such a tensor product is compatible with the canonical tensor product involution. As a consequence, we determine the forms of a tensor product of octonion algebras. Furthermore, we show that any such algebra does not satisfy the Skolem᎐Noether Theorem. Our results of Section 3 arise from a study of the generalized alternative nucleus of an algebra, since a tensor product of octonion algebras is generated by its generalized alternative nucleus. In Section 4, using -theoretic techniques, we classify finite dimensional simple unital algebras over an algebraically closed field of 0 which are generated by their generalized alternative nucleus, proving that such an algebra is the tensor product of a simple and a symmetric tensor product of octonion algebras. This result is used in Section 5 to sketch a variation of the Allison᎐Smirnov proof of the classification of finite dimensional central simple structurable algebras over a field of characteristic 0. Finally, in Section 6, we prove that if A is generated by its generalized s alternative nucleus, then the associated Ž.x, y trace ŽLLxy . is associative. Let F be a field and C a unital, nonassociative F-algebra. Then S is a if there exists a nondegenerate n: C ª F such that nxŽи y .s nxny Ž.Ž.for all x, y g C. The form n is uniquely determined by these conditions and is called the of C. We will write s wx n nC . Composition algebras only exist in rank 1, 2, 4, or 8Ž see 17. . Those of rank 4 are called algebras and those of rank 8 octonion algebras. A composition algebra C has a canonical in¨olution ␥ given by ␥ s y ª s Ž.x tx Ž.1C x, where the trace map t: C F is given by tx Ž. nŽ.1, x . An example of an eight-dimensional composition algebra is Zorn’s algebra of vector matrices ZorŽ.ŽF seewx 22, p. 507 for the definition. . The norm form of ZorŽ.F is given by the determinant and is a hyperbolic form. Composition algebras are quadratic. That is, they satisfy the identities

2y q s g x txxŽ.nx Ž.1C 0 for all x C, s nŽ.1C 1 ON THE TENSOR PRODUCT OF COMPOSITION ALGEBRAS 43 and are alternati¨e algebras; i.e., xy 2 s Ž.xy y and xy2 s xxyŽ.for all g s ␥ s ␥ s ␥ q x, y C. In particular, nxŽ. Ž.xx x Ž.x and tx Ž.1C Ž.x x. F For any composition algebra D over F with dim F Ž.D 4 and any ␮ g F=, the F- D [ D becomes a composition algebra via the multiplication

Ž.Ž.u, ¨ uЈ, ¨ Ј s Ž.uuЈ q ␮␥ Ž.¨ Ј ¨ , ¨ Јu q ¨␥ Ž.uЈ for all u, ¨, uЈ, ¨ Ј g D, with norm

¨ s y ␮ ¨ nuŽ.Ž., nuDD Ž.n Ž..

This algebra is denoted by CayŽ.D, ␮ . Note that the embedding of D into the first summand of CayŽ.D, ␮ is an algebra monomorphism. The norm ␮ y␮ m form of CayŽ.D, is obviously isometric to ² 1, :nD. Since two composition algebras are isomorphic if and only if their norm forms are isometric, we see that if C is a composition algebra whose norm form ( y␮ m ( ␮ satisfies nCD²:1, n for some D then C Cay Ž.D, . In particu- ( y m lar, ZorŽ.F Cay ŽD, 1 . for any D since² 1, 1 : nD is hyperbolic. A composition algebra is split if it contains an isomorphic copy of F [ F as a composition subalgebra, which is the case if and only if it contains zero divisors. Over algebraically closed fields any composition algebra of dimension G 2 is split.

2. ALBERT FORMS

From now on we consider only the fields F with charŽ.F / 2 unless stated otherwise. It is well known that any norm of a composition algebra is a 3-fold , and conversely any 3-fold Pfister form is the norm of some composition algebra. H Let C be a composition algebra. Define CЈ s ²:F1 s Ä x g C : txŽ.s < nxŽ.,1 s 04 . Then nЈ s n CЈ is the pure norm of C. Note that

Ј s g s f 2 g C Ä4x C : x 0or x F1CC and x F1 s Ä4x g C : ␥ Ž.x syx .

Moreover, C is split if and only if its norm n is hyperbolic, two composi- tion algebras are isomorphic if and only if their norms are isometric, and C is a if and only if n is anisotropic. We first investigate tensor products of two composition algebras. Follow- s m ing Albert, we associate to the tensor product C C1 F C2 of two s s q y composition algebras with dimŽ.Ciir and n Cin the Ž.r12r 2- X Hy X i y dimensional form n12²:1 n of determinant 1. This definition, for 44 MORANDI, PEREZ´¨-IZQUIERDO, AND PUMPLUN

wx C12or C an , was first given by Allison in 5 . In the Witt y WFŽ., obviously this form is equivalent to n12n . Like the norm form of a composition algebra, this Albert form contains crucial informa- tion about the tensor product algebra C. For biquaternion algebras, this is well-knownwx 1, Theorem 3; 19, Theorem 3.12 . We introduce some notation and terminology. If q is a quadratic form and if ވ s ²:1, y1 is the s H ވ hyperbolic plane, then q q00i for some anisotropic form q and i. The integer i is called the Witt index of q and is denoted by iqW Ž.. In the proof of the following proposition, we use the notion of linkage of Pfister formsŽ seewx 12, Section 4. . Recall that two n-fold Pfister forms q12and q are r-linked if there is an r-fold Pfister form h with s m X X q1 h qiifor some Pfister forms q . Finally, we call a two-dimensional commutative F-algebra that is separable over F a quadratic´ etale algebra. Note that any quadratic´ etale algebra either is a quadratic field extension of F or is isomorphic to F [ F. Part of the following result has been proved inwx 15, Theorem 5.1 . ¨ PROPOSITION 2.1. Let C12 and C be octonion algebras o er F with norms s s X H n12 and n , and let i iW Ž. N be the Witt index of the Albert form N n1 y X ²:1 n2 . s m Ž.i i 0 C12 and C do not contain isomorphic quadratic´ etale subalgebras. s m Ž.ii i 1 C12 and C contain isomorphic quadratic´ etale subalge- bras, but no isomorphic quaternion subalgebras. s m Ž.iii i 3 C12 and C contain isomorphic quaternion subalgebras, but C12 and C are not isomorphic. s m ( Ž.iv i 7 C12C . wxHy Proof. By 12, Propositions 4.4 and 4.5 , the Witt index of n12²:1 n r Hy is 2 , where r is the linkage of n12²:1 n . Note that the Witt Hy index of N is one less than the Witt index of n12²:1 n since Hy s ވ H ( ( s n12²:1 n N.IfC 1212C , then n n ,soi 7. Conversely, s Hy ( if i 7, then n12²:1 n is hyperbolic, so n12n , which forces ( C12C .IfC 1and C 2are not isomorphic but contain a common quater- s ␮ s m nion algebra Q, then CiiCayŽ.Q, for some i. Therefore, n1 nQ y␮ s m ␮ ²:1, 12and n nQ ²:1, 212. These descriptions show that n and n s s are 2-linked, so i 3. Conversely, if i 3, then n12and n are 2-linked but not isometric. If ²²a, b :: is a factor of both n12and n , then s s g = s n12²²a, b, c :: and n ²²a, b, d :: for some c, d F .IfQ y y s y y Ž.a, b , we get C11Cay Ž.Ž.Q, c and Cay Q, d ,soC and C2con- tain a common quaternion algebra. If C12and C contain a common quadratic´ etale algebra FtwxrŽt 2 y a. but no common quaternion algebra, ON THE TENSOR PRODUCT OF COMPOSITION ALGEBRAS 45

y then²: 1, a is a factor of n12and n , which means they are 1-linked. If n1 and n2 are 2-linked, then the previous step shows that C12and C have a common quaternion subalgebra, which is false. Conversely, if n12and n are 1-linked but not 2-linked, then C12and C do not have a common quaternion subalgebra, and if ²²a :: is a common factor to n12and n , then wxr 2 y C12and C both contain the´ etale algebra Ft Ž.t a . ¨ PROPOSITION 2.2. Let C12 be an octonion algebra o er F and C be a ¨ quaternion algebra o er F, with norms n12 and n . Again consider the Witt s X Hy X index i of the Albert form N n12²:1 n . s m Ž.i i 0 C12 and C do not contain isomorphic quadratic´ etale subalgebras. s m Ž.ii i 1 C12 and C contain isomorphic quadratic´ etale subalge- bras, but C21 is not a quaternion subalgebra of C . s m ( ␮ ␮ g = Ž.iii i 3 C12CayŽ.C , for a suitable F and C2 is a di¨ision algebra. s m ( ( Ž.iv i 5 C12ZorŽ.F and C MF2Ž..

Proof. In the case that both algebras C12and C are division algebras, wx this is an immediate consequence of 15, Lemma 3.2 . If both C12and C are split, then clearly N has Witt index 5. If C2 is a division algebra and s ␮ ␮ H ވ s m C12CayŽ.C , for some , then n2is anisotropic and N n2 y␮ Hy s ވ Hy␮ ²:²:1, 1 n224 ²:n ,so N has Witt index 3. Note that s the converse is easy, since if i 3 then n2 is isomorphic to a subform of s m n12, which forces n to be a factor of n 1.If n12²:1, a n , then ( y C12CayŽ.C , a ,soC 2is a subalgebra of C112.IfC and C contain a wxr 2 y common quadratic´ etale algebra Ft Žt a. but C2 is not a quaternion subalgebra of C112, then n and n have ²²a :: as a common factor, so s g s i 1. Finally, if N is isotropic, there are xiiC , both skew, with nx11Ž. s s wx wx nx22Ž.. Then, as tx11 Ž.0 tx 22 Ž., the algebras Fx1and Fx2are isomorphic, so C12and C share a common quadratic´ etale subalgebra. This finishes the proof. ( m If C is a biquaternion algebraŽ i.e., C C1 F C2 for two quaternion X Hy X algebras C12and C ., then the Albert form n1²:1 n 2is determined up to similarity by the isomorphism class of the algebra C wx19, Theorem 3.12 . Allison generalizes this resultwx 5, Theorem 5.4 to tensor products of arbitrary composition algebras. However, he always considers the involu- ␴ s ␥ m ␥ s m tion 12as a crucial part of the algebra C C1F C2 . Allison m ␥ m ␥ m ␥ m ␥ proves that Ž.C1 F C21, 2and Ž.C 3F C43, 4are isotropic algebras if and only if they have similar Albert forms, for the cases that

C13,C are octonion and C2, C 4are quaternion or octonion algebras. 46 MORANDI, PEREZ´¨-IZQUIERDO, AND PUMPLUN

␸ m ␥ m ␥ ª The fact that any F-algebra isomorphism : Ž.ŽC1 F C21, 2C 3 m ␥ m ␥ F C43, 4. between arbitrary products of composition algebras yields X Hy X ( ␮ X Hy X ␮ g = an isometry n1234²:1 n Žn ²:1 n . for a suitable F is s m = ª easy to see. Also, since for C C1 F C2 the map²: , : C C F m m s m given by ²:Ž.Ž.x1212x , y y nx 111222, y nx, y is a nondegenerate symmetric bilinear form on C such that ²Ž.␴ x , ␴ Ž.:²y s x, y :, the equa- tion ²zx, y :s ²x, ␴ Ž.:zy holds Ž that is, ² , : is an in¨ariant form; cf.w 6, p. 144x or Section 6 below.Ž.²Ž.: and ␶ : C = C ª k, ␶ x, y s x, ␴ y is an associative nondegenerate symmetric bilinear form which is proper, it m ( m follows easily that n1234n n n . Suppose that we have two algebras that each are a tensor product of an octonion algebra and a quaternion algebra. We obtain a necessary and sufficient condition for when their Albert forms are similar. We use the notation DqŽ.to denote the elements of F= represented by a quadratic form q.

THEOREM 2.3. Let C12, C be octonion algebras and Q1, Q 2 quaternion ¨ m algebras o er F. Let N12 and N be the Albert forms of C1F Q1 and m ¨ ( ␮ ␮ g = ( C2 F Q21212, respecti ely. If N N for some F , then Q Q . Moreo¨er, there is a quaternion algebra Q and elements c, d g F= such that ( ( ␮ ( y␮ g C12CayŽ.Q, c , C Cay ŽQ, d ., Cay ŽQ 1, .Cay ŽQ, cd ., and c ¨ ␮ g DnŽ.C . Con ersely, if there is a quaternion algebra Q and elements c, d, = 2 s ¨ F such that C11, Q Q 2, and C 2 satisfy the conditions of the pre ious ( ␮ sentence, then N12N . ( ␮ ␮ g = ª Proof. Suppose that N12N for some F .If c: WFŽ.Br Ž.F s ␮ s is the Clifford invariant, then cNŽ.122c ŽN .cN Ž.. Since c is trivial on IF3Ž., we have cN Ž .s c Žyn .and cN Ž .s c Žyn .Žseew 23, Chap. 1 Q122 Q 5.3x.Ž.Ž. . Therefore, cn s cn . However, the Clifford invariant of the QQ12 norm form of a quaternion algebra is the class of the quaternion algebra, wx s wx ␮ s wx by 23, Corollary V.3.3 . This implies that cNŽ.11Q and cŽ.N 22Q . Since wxQ s wxQ , we get Q ( Q . As a consequence of this, n ( n . 12 12 QQ12 Thus,

n (y²:1,y␮ m n ( n Hyn H ␮n CQ11C1Ž.Q1Q1 ( ␮ n Hyn H ␮n ( ␮n H ␮n Hy␮n Ž.CQ22 QC 12 Ž Q 1 Q 2 . ( ␮n H 4ވ. C 2

The forms n and²: 1, y␮ m n are Pfister forms. The line above shows CQ11 that these Pfister forms are 2-linked, in the terminology ofwx 12 . Therefore, there is a 2-fold Pfister form ²²y a, yb ::with n (y ²²a, yb, yc :: C1 and² 1, y␮ :m n (y ²² a, yb, ye:: for some c, e g F=. An elemen- Q1 ON THE TENSOR PRODUCT OF COMPOSITION ALGEBRAS 47

tary calculation shows that

²²ya, yb, yc ::Hy ²² ya, yb, ye :: ( 4ވ Hy²c, e :my ²²a, yb ::. ␮ (y myy (y␮ y Therefore, nCC²c, e : ²²a, b ::. Thus, n c²² y y 2 y y y 2 a, b, ce::. Since nC and ²² a, b, ce:: are Pfister forms, we get (yy y 2 s nC ²² a, b, ce::. If we set d ce and let Q be the quaternion 2 (yy y ( algebra Ž.a, b FC, then the isomorphisms n ²² a, b, c:: and n C y y y ( 12( ²² a, b, d:: give C12CayŽ.Q, c and C CayŽ.Q, d . Moreover, (y␮ y␮ g y␮ m nCCcn ,so c DnŽ.C. Finally, the isomorphism² 1, : 22(yy y 2␮ ( ( nQ ²² a, b, e::gives Cay ŽQ1, .Cay ŽQ, e .Cay ŽQ, cd .. 2 s s It is a short calculation to show that if C12CayŽ.Q, c , C Cay ŽQ, d ., and Q s Q is a quaternion algebra with CayŽ.Q , ␮ ( Cay ŽQ, ydc ., XXXX12 1 then n Hy²:1 n ( ␮ Žn Hy ²:.1 n . CQCQ1122 The argument of the previous theorem does not work for a tensor product of two octonion algebras since the Albert form is an element of IF3Ž., whose Clifford invariant is trivial.

COROLLARY 2.4. With the notation in the pre¨ious theorem, suppose that ( ␮ ␮ g = N12N for some F . If one of C1 and C 2 is split, then the other ␮ algebra is isomorphic to CayŽ.Q1, . Proof. We saw in the proof of the previous proposition that

n Hy²:1,y␮ m n ( ␮n H 4ވ. CQ11C2

Suppose that C is split. Then n Hy²:1, y␮ m n is hyperbolic, so 2 CQ11 n ( ²:1, y␮ m n . Therefore, C ( CayŽ.Q , ␮ . On the other hand, if CQ1211 C is split, then n ( 4ވ, so by Witt cancellation y␮n ( ²:1, y␮ m 1 CC1 2 n . Since n and²: 1, y␮ m n are both Pfister forms, this implies that QC12 Q1 n ( ²:1, y␮ m n , and so C ( CayŽ.Q , ␮ . CQ2121 ( ␮ ( In Theorem 2.3 above, it is possible for N12N without C 12C . Moreover, the quaternion algebra Q of the proposition need not be

isomorphic to Q1. We verify both of these claims in the following example.

EXAMPLE 2.5. In this example we produce nonisomorphic octonion

algebras C12and C and a quaternion algebra Q1that is not isomorphic to a subalgebra of either C12or C and is such that the Albert forms of m m C1 F Q12and C F Q1 are similar. To do this we produce nonisometric Pfister forms ²²x, y, z ::and ²²x, y, w :: and elements u, ¨, ␮ with ²²x, y, zw ::( ²²u, ¨, ␮ ::such that the Witt indexes of ²²x, y, z :: H y²²u, ¨, ␮ ::and ²²x, y, w ::Hy ²²u, ¨, ␮ :: are both 2 and ␮ z g sy y s y s DxŽ²², y, w ::.. We then set Q Žx, y ., C12Cay ŽQ, z ., C 48 MORANDI, PEREZ´¨-IZQUIERDO, AND PUMPLUN

y sy y¨ ( ␮ CayŽ.Q, w , and Q11 Žu, .. From Theorem 2.3, we have N N2. However, Proposition 2.2 shows that Q1 is not isomorphic to a subalgebra of either C12or C . Moreover, C1and C 2are not isomorphic since their norm forms are not isometric. Note that Q and C2 are not isomorphic since Q11is not a subalgebra of C . Let k be a field of characteristic not 2, and let F s kxŽ., y, z, w be the ␮ s s rational function field in four variables over k. Set xyzw, n1 s ²²x, y, z ::, and n2 ²²x, y, w ::. By embedding F in the Laurent series s H field kxŽ, y, zw .ŽŽ .., we see that n12and n ²²x, y ::wx ²², y :: are not isomorphic over this field by Springer’s theoremw 23, Proposition x ␮ s 2 VI.1.9 , so n12and n are not isomorphic over F. Also, z zŽ. xyw , sy y which is clearly represented by n21. Set Q Ž.zw, xzw . A short calcu- lation shows that ²²x, y, zw ::s ²²zw, xzw, ␮ ::. Finally, for the Witt indices, we have

n Hyn s ²:²:1, x, y, xy, z, xz, yz, xyz Hy 1, zw, xzw, x 1 Q1 s 2ވ H ²:y, xy, z, xz, yz, xyz, yzw, yxzw s 2ވ H ²:²:y, xy, z, xz, yz, xyz H w y z, yxz .

The Springer theorem shows that this form has Witt index 2. Similarly,

n Hyn s ²:²:1, x, y, xy, w, xw, yw, xyw Hy 1, zw, xzw, x 2 Q1 s 2ވ H ²:y, xy, w, xw, yw, xyw, yzw, yxzw s 2ވ H ²:²y, xy H w 1, x, y, xy, yz, yxz : has Witt index 2. For the remainder of this section we will also consider the case that s charŽ.F 2. Let C12and C be composition algebras over F of dim Fi ŽC . s G wx rii2, and let n be the norm form of Ci. Using the notation of 21 , the s s m y m s subspace QCŽ.12, C Äu x1 1 1 x211: txŽ.tx 22 Ž.4 has dimen- q y s y ␥ m ␥ g m sion r12r 2, and QCŽ.12, C Ä z Ž.Ž.1 2z : z C 1F C24 is the m ␥ m ␥ set of alternating elements of C1 F C212with respect to . The ª m y nondegenerate quadratic form N: QCŽ.12, C F given by Nx Ž1 1 1 m s y X Hy X x21122.Ž.Ž.nx nx is isometric to the Albert form n1²:1 n 2of m C1 F C2 . q y ; ސ r1 r 2 3 Let VN be the F-quadric defined via N. In the case that / charŽ.F 2, VNNcoincides with the open subvariety U of closed points m y m f f w x12121 1 x with x F1 and x F1. We now generalize 21, Propositionx in the following two propositions. We will make use of the wx following fact that comes from Galois theory: Let Fzi be the commuta- g s tive F-subalgebra of dimension two of Ciigenerated by z Cifor i 1, 2. ON THE TENSOR PRODUCT OF COMPOSITION ALGEBRAS 49

␣ wxª; w ␣ s Then there exists an isomorphism : Fz12Fz4 such that Ž.z12z s s if and only if nz11Ž.nz 2 Ž. 2and tz 11 Ž.tz 2 Ž. 2.

PROPOSITION 2.6. There exists a bijection ⌽ between the set of F-rational ␣ points of UN and the set of triplesŽ. K12, K , , where Ki is a two-dimensional ¨ ␣ ª; commutati e subalgebra of Ci and where : K12KisanF-algebra isomorphism,

⌽ g ª; ␣ ␣ ¨ : Ä4P UN : PanF-rational point Ä4Ž.K12, K , : K 12, K , as abo e wx wx␣ wxª; wx s m y m ¬ Fz12, Fz , : Fz 1Fz 2 P z121 1 z ¬ . ž/z12z g Proof. Any F-rational point P UN corresponds with an element m y m g s s x1 1 1 x2QCŽ 1, C 2 .with tx 11 Ž.tx 2 Ž. 2and nx 11 Ž.nx 2 Ž. 2. ␣ wxª; wx ¬ Then there exists an F-algebra isomorphism : Fx121Fx with x m y m s m y m x21. For x 1 1 x 21z 1 1 z 2it can easily be verified that

wx wx␣ wxª; wx Fx12, Fx , : Fx 1Fx 2 ¬ ž/x12x wx wx␤ wxª; wx s Fz12, Fz , : Fz 1Fz 2 ¬ . ž/z12z

Therefore, the mapping ⌽ is well defined. ␣ g X s Given a triple Ž.K12, K , , there are elements ziiC such that K i wx␣ wxª; w x ¬ Fzi and : Fz12Fz with z 12z . By the remark before the s s m y proposition, we have nz11Ž.nz 2 Ž. 2and tz 11 Ž.tz 2 Ž. 2; thus Nz Ž1 1 m s g 1 z2 . 0 and the triple defines the F-rational point P UN corre- m y m ⌽ sponding to z121 1 z .So is surjective. ⌽ m y m s ⌽ m y To prove injectivity, suppose that Ž.Žx1211 1 x z 1 1 m wxs wx wxs wx␣ wxª; z21122.. Then Fx Fz , Fx Fz , and the maps : Fx1 wx ¬ ␤ wxª; wx ¬ wxs Fx21, x x 2, and : Fz1Fz 2, z z 2, are equal. Since Fxi wx s q s q g Fzi , we write x1122a bz and x c dz with a, b, c, d F.We s s have a c since tx11Ž.tx 2 Ž. 2. Therefore, we may replace x1with bz 1 m y m m y m and x22with dz without changing x11 1 x 2. Thus, x 11 1 s m y m s s x21bz 1 1 dz2, and nx 11221122Ž.nx Ž., nz Ž.nz Ž.imply that s 2 s 2 22s s " nx11Ž. bn 11Ž. z and nx 2 Ž. 2dn 22Ž. z . Therefore, b d ,sob d. s ␣ s ␣ s s m y m Now x21Ž.x Žbz 12 . bz yields b d, and we get x11 1 s m y m ⌽ x21bzŽ.1 1 z 2which shows that is injective. s s m y m f In the case that charŽ.F 2, the set UN Ä x1211 1 x : x F1, f x2 F14 is a proper open subvariety of VN . The proof of the previous proposition shows that ⌽ again is a bijection between the F-rational points 50 MORANDI, PEREZ´¨-IZQUIERDO, AND PUMPLUN

␣ of UN and the triples Ž.K12, K , , where K i is a two-dimensional commu- ␣ ª; tative F-subalgebra of Ci and : K L is an F-algebra isomorphism. We can say more in this situation.

s PROPOSITION 2.7. Let charŽ.F 2. There exists an F-rational point in VN ␣ if and only if there exists a tripleŽ. K12, K , such that Ki is a quadratic´ etale ␣ ª; subalgebra of Ci and : K12KisanF-algebra isomorphism. In addition, l s there exists an F-rational point in VN Ätx11Ž. 04 if and only if there exists a ␣ tripleŽ. K12, K , such that K1 and K 2 are purely inseparable quadratic ␣ ª; extensions and : K12KisanF-algebra isomorphism.

Proof. As pointed out before the proposition, there is a bijection ␣ : between F-rational points in UN and triples Ž.K12, K , with K iiC commutative subalgebras of dimension 2 over F. To prove the first statement, only one half needs further argument. Suppose VN has an F-rational point. Since VNNis a quadric hypersurface, V is then bira- q y tionally equivalent to ސ r1 r 2 3 . The F-rational points of projective space are dense, so there is an F-rational point in UN . Therefore, we get a triple ␣ Ž.K12, K , with K iia quadratic´ etale subalgebra of C . l s For the second statement, an F-rational point in VN Ätx11Ž. 04 m y m s corresponds with an element x1211 1 x such that nxŽ.1nx2 Ž.2 s s wx and tx11Ž.tx 2 Ž. 2 0, so Fxi is a purely inseparable extension. There ␣ wxª; wx ␣ s exists an isomorphism : Fx12Fx with Ž.x 12x and thus a triple wx wx␣ ␣ ŽFx12, Fx , .Ž.. Conversely, if there is a triple K12, K , with K i g s s wx purely inseparable, there are xiiC with tx iiŽ. 0 such that K Fx1 , s wx␣ wxª; wx ¬ s m L Fx212, and : Fx Fx , x1x2,sonx1Ž.1nx2 Ž.21and x y m l s 1 1 x2 defines an F-rational point in VN Ätx11Ž. 0.4

3. THE AUTOMORPHISM GROUP OF A TENSOR PRODUCT OF OCTONION ALGEBRAS

In this section we compute the automorphism group, the derivation algebra, and the forms of a tensor product of a finite number of octonion / s m иии m algebras over a field F with charŽ.F 2. Let C C1 FFnC be the m tensor product of octonion algebras. As we will see, the subspace C1 F F m иии m q иии q m иии m m FFF F FFFnF C is responsible for many proper- ties of C, so our first goal is to characterize it. Recall that the associati¨e nucleus of an algebra A is NŽ.A s Äa g A : Ž.Ž.Ž.a, A, A s A, a, A s A, A, a s 04 , where Ž.Ž.x, y, z s xy z y xyzŽ.denotes the usual associator. The commutati¨e nucleus of A is the set KŽ.A s Äa g A : wŽa, Ax s 0.4 ON THE TENSOR PRODUCT OF COMPOSITION ALGEBRAS 51

DEFINITION 3.1. The subspace

s g sy s ᭙ g Nalt Ž.A : Ä4a A : Ža, x, y . Žx, a, y . Žx, y, a . x, y A will be called the generalized alternative nucleus of A. Remark 3.2. The alternati¨e nucleus was introduced by Thedywx 35 as

Ä a g A : Ž.x, a, x s 0 and Ž.Ž.Ž.a, x, y s x, y, a s y, a, x for all x, y g A4 and is a subalgebra of A. The generalized alternative nucleus differs from this nucleus and, in general, it may not be closed under products, although it possesses an interesting algebraic structureŽ. see Proposition 4.6 . s s PROPOSITION 3.3. Let A12, A be unital algebras with NŽ.A1F s s m s m q KŽ.Aor22N Ž.A F K Ž.A 1a. Then N lt1 ŽA F A2alt1 .N Ž.A F F m F F NaltŽ.A 2 . s s Proof. By symmetry we can assume that NŽ.A12F K Ž.A . Let s m X g m X a Ýaiia NaltŽ.A 1 F A2 with ai linearly independent. The identi- ties defining the generalized alternative nucleus with x replaced with m s m g x 1 and y with y y 1 show that ai NaltŽ.A 1 . A similar argument m : with ai linearly independentŽŽ.. in NaltA 1shows that N alt ŽA 1 F A2 . m sy NaltŽ.Ž.A 1 F NaltA 2 . Now the identity Ž.Ž.a, x, y x, a, y with x re- m Ј m m X Ј s placed with x x and y with y 1 leads to ÝŽ.aii, x, y ax y m Ј X g w m ÝŽ.x, aiii, y x a . Since a NaltŽ.A 1 , this implies that ÝŽ.ai, x, y X m Јx s m ai,1 x 0. But, by hypothesis, the centralizer of A21in A F A2 is m m XXg m s X A1 FiiF, hence ÝŽ.a , x, y a A1 F F. By choosing a1 1 and ai s G g linearly independent, we get that Ž.aii, x, y 0if i 2, and since a s s g NaltŽ.A 1 , it follows that Ž.Ž.x, aii, y 0 x, y, a , too. Therefore, a i s G m ; m q m NŽ.A1aF if i 2 and N ltŽA1F A2alt1 .N Ž.A FFF F NaltŽ.A 2 . The other inclusion is obvious. m иии m In general, in a tensor product of algebras A1 FFnA we will m иии m m иии m identify the factors AiFwith the subalgebra F FAiFFF m иии m s without mention; thus, for instance, we will write A1 FFnA Ł iiA . m иии m s q иии q COROLLARY 3.4. NaltŽ.C 1 FFnC C1 Cn. wx s s Proof. It is well known 37, p. 41 that NŽ.CiiF K Ž.C and that m s m m s m NŽ.Ž.Ž.Ž.Ž.Ž.A1 F A21N A F N A21and K A F A21K A F K A2 . 52 MORANDI, PEREZ´¨-IZQUIERDO, AND PUMPLUN

Recall that in any algebra with product denoted bywx , the element JxŽ., y, z s wwx, y x, z xq wwy, z x, x xq wwz, x x, y x is called the Jacobian of x, y, z. The algebra is called a Malce¨ algebra if it is anticommutative and JxŽ , y, wxx, z . s w JxŽ., y, z , xx. One important example of a simple Malcev algebra is the algebra of elements of zero trace in an octonion algebra with wx the product given by the commutator 24, 26, 29, 30 . Therefore NaltŽ.C is a s [ XXX[ иии [ Malcev algebra, and NaltŽ.C F1 C1 Cniwith C minimal ideals that are simple Malcev algebras and F1 the center. The derived algebra of

NaltŽ.C is

XXs s [ иии [ X Nalt Ž.C Nalt Ž.C ,N alt Ž.C C1 Cn .

␸ X ª X Remark 3.5. Let 01: C C 2be an isomorphism of Malcev algebras. ww xxsy q ␸ ␸ s Since a, a, b 4nab11Ž.2na Ž, ba ., we have n2 Ž00 Ž.a , Ž..b ␸ ª ␣ q ¬ ␣ q ␸ na11Ž., b , so we can define : C C20by 1 a 1 Ž.a , which is an isomorphism because of the identity 2 ab s wxa, b y naŽ., b . That is, X X 1 any isomorphism from C12onto C is the restriction of an isomorphism ␴ g between C12and C . Moreover, given an automorphism AutŽ.F then ␴ ␸ X X any -semilinear isomorphism 012between C and C is induced by a ␴-semilinear isomorphism ␸: ␣1 q a ¬ ␴␣Ž.1 q ␸ Ž.a between C and X 01 C21. In the same way, any derivation of C is the restriction of a derivation m иии m ␸ of C11. Something similar holds for C FFnC . Let 0 be an auto- X [ иии [ X X morphism of C1 Cni. Since C are the minimal ideals there exists a ␲ g ␸ X s X permutation Ýn such that 0ŽCi . C␲ Ži.. Therefore, by the previous, ␸ < X ␸ ª 0 C i is the restriction of an isomorphism ii: C C␲ Ži. and we can define ␸ m иии m an automorphism of C1 FFnC such that the restriction to C iis ␸ X [ иии [ X i. Hence, any automorphism of C1 Cn is the restriction of an m иии m ␴ automorphism of C1 FFnC . The same is true for -semilinear automorphisms.

PROPOSITION 3.6. The restriction map gi¨es the isomorphisms

( X AutŽ.C AutŽ. Nalt Ž.C , ( X ( [ иии [ DerŽ.C DerŽ. Nalt Ž.C Der ŽC1 .Der ŽCn ..

X Proof. Any automorphismŽ. resp. derivation of C leaves N Ž.C invari- altX ant, so it induces an automorphismŽ. resp. derivation of N Ž.C . Since X alt NaltŽ.C generates C as an algebra, the restriction map induces monomor- ª X ª X phisms AutŽ.C Aut Ž NaltŽ..C and Der Ž.C Der Ž NaltŽ..C . In the case of ON THE TENSOR PRODUCT OF COMPOSITION ALGEBRAS 53

AutŽ.C this monomorphism is also an epimorphism by Remark 3.5. In the g X case of DerŽ.C , given d Der Ž NaltŽ..C we have XXXXXXXXs wx: q : dCŽ.iiiiiiiidCŽ., C dCŽ., C C , dCŽ. C X s X [ иии [ X X so DerŽ NaltŽ..C Der ŽC1.ŽDer Cni.. Since any derivation d of Ci m m иии m q иии q is induced by a derivation diiof C , and d iid id id m иии m m X [ иии [ id dn is a derivation of C, it follows that DerŽC1. X s DerŽ.Cn Der Ž.C . ␴ Let be the tensor product of the canonical involutions of the Ci and AutŽ.C, ␴ the automorphisms of C that commute with ␴ .

COROLLARY 3.7. With the pre¨ious notation, AutŽ.c s Aut ŽC, ␴ .. m m m ( n1 m иии m n m n i Remark 3.8. We can write C C1 FFmC with C ithe \ / q иии q s tensor product of niiijcopies of C and C C if i j and n1 nm X ( XXX[ иии [ n. Thus, NaltŽ.C nC1 1 nCmmwith nC iiisomorphic to the direct XXX\ / X sum of niiijcopies of C , and C C if i j. Since AutŽnCii. is the wreath X m product of AutŽC .Žand the symmetric group Ý , we obtain that Aut C n i . ini ( AutŽ.C , the wreath product of AutŽ.C and Ý , and thus AutŽ.C ( ini i ni AutŽ.C = иии = Aut ŽC .. 1 nm1 nm X ( XX[ иии [ The uniqueness of the decomposition NaltŽ.C C1 Cn gives the following uniqueness of the factorization of C. m иии m PROPOSITION 3.9. Let A12, A be unital algebras such that C1FF ( m s ⌳ j ⌳ Cn A1 F A212. Then there exists a partition Ä41,...,n such that ( m ( m Ä4 A1 ig ⌳ Ci and A2 jg ⌳ Cj. In particular, the factors C1,...,Cinn m иии m12 C1 FFnC are uniquely determined up to order and isomorphism. s m Proof. Since the associative and commutative nuclei of C C1 F иии m s s s FnC are each the base field, NŽ.Ai F K Ž.Aifor i 1, 2. By Proposition 3.3, mq m ( q иии q s [ X [ иии [ X NaltŽ.A 1 FFF NaltŽ.A 2C 1 Cn F1 C1 Cn . X s The Ci are minimal ideals; therefore, there exists a partitionÄ4 1, . . . , n ⌳ j ⌳ ( [ [ X m 12such that NaltŽ.AijF1 g ⌳ C . Thus, the image in A1 F j i X A of the subalgebra mg ⌳ C generated by [g ⌳ C is contained in A . 2 j iijjij Since the two algebras have the same dimension, they must be equal. m иии m m иии m COROLLARY 3.10. Two algebras C1 FFnC and C˜˜1 FFmC are isomorphic if and only if n s m and there exists a permutation ␶ such that

Ci is isomorphic to C˜␶ Ži..

EXAMPLE 3.11. It is well known that if the Albert forms of two biquaternion algebras are similar, then the algebras are isomorphic. We 54 MORANDI, PEREZ´¨-IZQUIERDO, AND PUMPLUN

give an example to show that the analogue of this result is false for

octonion algebras. Let C12and C be nonisomorphic octonion F-algebras m m and consider the tensor products C1 F C12and C F C2 . Then their X Hy X X Hy X Albert forms are nCCn and n CCn , respectively. Therefore, 11 22 m these forms are isomorphic as they are both hyperbolic. However, C1 F C1 m is not isomorphic to C2 F C21by the previous corollary since C and C2 are not isomorphic. The following result points out the special role played by the involu- tion ␴ . ¨ m иии m COROLLARY 3.12. The only in olution of C1 FFnC which com- mutes with all automorphisms is ␴ . ␴ Ј s m иии m Proof. Let be another involution of C C1 FFnC commut- : ing with AutŽ.C . The elements fixed by AutŽCi .Aut Ž.C are exactly sm ␴ Ј s Cˆˆij/ ijC . Therefore, Ž.C iCˆ i. Looking at the centralizer of C ˆ i ␴ Ј s ␴ Ј␴ yields Ž.CiiC . The automorphism induces an automorphism of ␴ Ј␴ s wx ␴ Ј s ␴ Ciiwhich commutes with AutŽ.C . That is, id 17 and . ␴ ¨ m иии m We will call the canonical in olution of C1 FFnC . ␸ s m иии m ª s m иии m COROLLARY 3.13. Let : C C1 FFnC C˜˜C1 FFnC ˜ be an isomorphism. If ␴ and ␴ Ј are the canonical in¨olutions of C and C˜ respecti¨ely, then ␸␴ s ␴ Ј␸. Proof. Since AutŽ.C˜ s ␸ Aut Ž.C ␸y1, then ␸␴␸y1 commutes with AutŽ.C˜ . Therefore, ␴ Ј s ␸␴␸y1. We now show that the Skolem-Noether theorem does not hold for C.

COROLLARY 3.14. There exist simple F-subalgebras B and BЈ of C and an F-algebra isomorphism f: B ª BЈ such that there is no F-algebra automor- < phism ␸ of C with ␸ B s f. s Proof. Let Qiibe a quaternion subalgebra of C for i 1,2 and let f s m be an F-algebra automorphism of A Q1 F Q2 that is not compatible < with ␴ A; such maps exist since we can take f to be the inner automor- phism of an element t g A with ␴ Ž.ttf F. The condition ␴ Ž.ttg F is < precisely the condition needed to ensure that f is compatible with ␴ A. s q g s m For example, we can take t 1 ii12 A Q1 F Q2 Žwhere the stan- ␸ dard generators of Qrrare i and j r..If f extends to an automorphism < of C, then ␸Ž.A s A,so␸ is compatible with ␴ . This forces ␸ A s f to < be compatible with ␴ A, and f is chosen so that this does not happen. We devote the remainder of this section to computing the forms of s m tensor products of , that is, F-algebras A such that AKFK ON THE TENSOR PRODUCT OF COMPOSITION ALGEBRAS 55

( m иии m r A C1 KKnC for some extension K F and octonion algebras Ci over K. We will denote C m иии m C by T. X 1 XKKnXX Since K m N Ž.A ( N Ž.K m A ( C [ иии [ C , the algebra X F alt alt F 1 n NaltŽ.A is separable. It is worth noting that for a finite dimensional r m ( separable F-algebra R and a field extension K F such that K F R R1 [ иии [ Rni, with R central simple K-algebras, there exists a subfield K 0 r m ( of K such that K 00F is a finite Galois extension and K F R R˜1 [ иии [ R˜˜with R central simple K -algebras and R ( K m R˜. ni 0 iKi0 LEMMA 3.15. Let A be a form of a tensor product of octonion algebras ¨ : : o er F. There exists a finite Galois extension F K 0 K such that AK is the ¨ 0 tensor product of octonion algebras o er K 0.

Proof. Let K 0 be a finite Galois extension of F contained in K such m X s X [ иии [ X that K 0 F NaltŽ.A C˜ 1 C˜niwith C˜ octonion algebras over K 0 m XX( and K KiC˜ C i. By Remark 3.5, this isomorphism is induced by an 0 m ( m isomorphism K KiC˜˜C i. Thus we have an isomorphism K KŽC1 m иии m ( 0 m иии m0 KKnC˜˜. T which restricts to an isomorphism C1 KKnC˜ ( 00m 00 K 0 F A. This proposition allows us to assume in the following that KrF is a finite Galois extension. We denote the F-subalgebra generated by S by alg FF²:S and the subspace spanned by S by span ²:S . Since

m XX( m s XX[ иии [ K FKalg²:² NaltŽ.A algF Nalt ŽK FKA . :alg ²C1 Cn: s s m T K F A,

X s X we obtain A from NaltŽ.A since A algF ² NaltŽ.:A .

PROPOSITION 3.16. The map

m иии m ª XX[ иии [ Ä4F-forms of C1 KKnC ÄF-forms of C1 Cn4 ¬ X A Nalt Ž.A is a bijection with in¨erse gi¨en by NЈ ¬ alg²:NЈ . Moreo¨er, if A and B are m иии m ( X ( X F-forms of C1 KKnC then A B if and only if NaltŽ.A N altŽ.B . X XX[ иии [ Proof. It is clear that NaltŽ.A is an F-form of C1 Cn if A is an m иии m Ј X F-form of C1 KKnC . Conversely, let N be an F-form of C1 [ иии [ X ␴ g r Cn and ÄU␴ : GalŽ.K F 4 the semilinear automorphisms of X [ иии [ X Ј C1 Cn such that N is the set of fixed elements. By Remark 3.5, we can assume that U␴ is the restriction of a ␴-semilinear automorphism U˜␴ m иии m ␴ g of C1 KKnC . The algebra A of fixed elements by ÄU˜␴ : r m иии m Ј GalŽ.K F 4 is an F-form of C1 KKnC containing N . In fact, since 56 MORANDI, PEREZ´¨-IZQUIERDO, AND PUMPLUN

Ј XX[ иии [ Ј s X Ј s N extends to C1 Cn, we get N NaltŽ.A , and thus alg F ²N : A m иии m is an F-form of C1 KKnC . ( X ( X ␸ X ª X If A B then NaltŽ.A NaltŽ.B . Conversely, if 0:N altŽ.A NaltŽ.B ␸ X [ иии [ X is an isomorphism, it induces an automorphism ˜01of C Cn that, ␸ m иии m by Remark 3.5, is the restriction of an automorphism ˜ of C1 KKnC . Since ␸ s ␸ X s ␸ XXs s ˜˜Ž.A Ž.Ž.Ž.algF ² NaltŽ.A : alg F²:²:˜NaltŽ.A algF NaltŽ.B B, it follows that A ( B. This proposition allows us to construct easily the forms of a tensor X s [ product of octonion algebras. First, observe that if NaltŽ.A N1N 2 then m ( [ X s Ä4s ⌳ j ⌳ K FiN jg ⌳ Cj, i 1, 2 for some partition 1, . . . , n 12.By i s ²: m Proposition 3.16, AiFialg N is an F-form of jg ⌳ Cjand hence ( m i A A1 F A2 . Therefore, it is enough to construct the forms of a tensor product of octonion algebras with simple generalized alternative nucleus. X Let A be an F-algebra with NaltŽ.A simple and K a finite Galois m ( m иии m extension such that K F A C1 KKnC for some octonion alge- X ⌫ s ⌫ X bras Ci over K. Since NaltŽ.A is simple, the centroid ŽN altŽ..A is a finite separable extension of F. In fact, m ⌫ ( ⌫ m X ( ⌫ X [ иии [ X ( [ иии [ K FFŽ.K NaltŽ.A ŽC 1 Cn . K K implies that ⌫ is an extension of degree n and that we have n different ␴ ⌫ ª ␴ ⌫ F-monomorphisms ii: K. Every allows us to define a right -vec- ␣ (␥ s ␣␴ ␥ ␣ g ␥ g ⌫ tor space structure on K by iŽ., K, . We denote this ␴ new vector space by K 1. Now, m X ( m ⌫ m X ( m ⌫ m X K F NaltŽ.A K F Ž.⌫ NaltŽ.A ŽK F .⌫ Nalt Ž.A ␴ ␴ X ( 1 [ иии [ n m Ž.ŽK K ⌫ Nalt A. ␴ X ␴ X ( 1 m [ иии [ n m Ž.Ž.K ⌫ NaltŽ.A K ⌫ NaltŽ.A ( X [ иии [ X C1 Cn

␴ X X i m ( implies that, up to order, K ⌫ NaltŽ.A Ci. Therefore, we can think of ⌫ X X w the -algebra NaltŽ.A as a form of Ci. By the arguments in 24, pp. ᎐ x X ( Ј 240 241 , for instance, we can conclude that NaltŽ.A C for some ␴ ⌫ m Ј ( 1 m octonion algebra C over . Under the isomorphism K F C Ž K ⌫ ␴ CЈ.Ž[ иии [ K n m⌫ CЈ.Žwe identify x g CЈ with 1 m x.q иии qŽ1 m x.. ␴ ␴ Since we can view Ž K 1 m⌫⌫CЈ.Ž[ иии [ K n m CЈ. as the generalized ␴ ␴ 1 m Ј m иии m n m Ј alternative nucleus of Ž K ⌫ C .ŽKKK ⌫ C ., the algebra A m m иии m m corresponds with the F-subalgebra generated byŽ. 1 ⌫ x KK Ž1 ⌫ q иии q m m иии m m ⌫ 1.Ž.1 ⌫ 1 KK Ž.1 ⌫ x . Conversely, given an octonion -al- ON THE TENSOR PRODUCT OF COMPOSITION ALGEBRAS 57 gebra C with ⌫ a finite separable extension of F, it is easy to check that the algebra A constructed as above is a form of a tensor product of octonion algebras with a simple generalized alternative nucleus.

4. THE GENERALIZED ALTERNATIVE NUCLEUS

The generalized alternative nucleus is responsible for many properties of the tensor product of octonion algebras. In this section we pay special attention to this nucleus. We classify the simple finite dimensional unital algebras which are generated by their generalized alternative nucleus. Our methods rely on the representation theory of some Lie algebras; therefore, in this section we will assume that charŽ.F s 0 and that F is algebraically closed. We make free use of Lie algebra terminology and refer the reader to the books of Humphreyswx 16 and Jacobson wx 18 for definitions and results. Let C be an octonion algebra over F and SymnŽ.C the symmetric m иии m tensors of C FFC, the tensor product of n copies of C. The m m иии m q иии q m иии m m g elements a 1 1 1 1 a with a C lie in NaltŽC m иии m n n FFC. and generate Sym Ž.C . Therefore Sym Ž.C is an algebra generated by its generalized alternative nucleus. However, Sym nŽ.C is no longer simple. The contraction SymnŽ.C ª Symny2 Ž.C induced by x m иии m x ¬ nxxŽ.m иии m x is an epimorphism whose nucleus we will G ᎐ denote by TnŽ.C , n 2. We recover the Kantor Smirnov structurable s wx algebra when n 2 33, 7 . We will see that TnŽ.C is a unital simple s s algebra generated by NaltŽŽ.. Tn C . We set T10 Ž.C C and T Ž.C F.We now give our classification result.

THEOREM 4.1. Any simple finite dimensional unital algebra o¨er an algebraically closed field of characteristic zero which is generated by its generalized alternati¨e nucleus is isomorphic to the tensor product of a simple ¨ associati e algebra and TnŽ.C for some n. Recall fromwx 28 that a ternary deri¨ation of an algebra A is a triple g = = Žd123, d , d .End FFF Ž.A End Ž.A End Ž.A such that s q dxy12Ž.dxy Ž.xd 3 Ž. y Ž.1 for any x, y g A. The Lie algebra of ternary derivations is denoted by s s TderŽ.A .If d123d d then Ž. 1 says that d1is a derivation, and in that ¨ s q case we will say that Ž.d123, d , d represents a deri ation. Let TaaaL R . It is worth noting that

g m y y g a Nalt Ž.A ŽLaa, T , L a .and ŽR a, R aa, T .Tder Ž.A .2 Ž. 58 MORANDI, PEREZ´¨-IZQUIERDO, AND PUMPLUN

The following identities will be useful. g g LEMMA 4.2. Let a, b NaltŽ.A and x A. Then s q wxs q wx Ž.i LaxLL a xR a, L x, L xaLL x aL x, R a . s q wxs q wx Ž.ii R axRR x aR x, L a, R xaRR a xL a, R x . wxwxs Ž.iii Lab, R R ab, L . wxs y wxwxsy y wx Ž.iv Lab, L Lw a, bx 2 R ab, L , R ab, R Rw a, bx 2 Lab, R . s wxwxwxq q ¨ Ž.v The map Da, babababL , L L , R R , R is a deri ation s y wx s q w x of A, Da, b adw a, bx 3 Lab, R and 2 D a, b adw a, bx adab , ad , where ¬ wx ada : x a, x . Proof. PartsŽ. i and Ž ii . follow from the identities Ž.Ž.a, x, y s x, y, a , Ž.Ž.Ž.Ž.Ž.Ž.x, a, y sy x, y, a , y, a, x sy a, y, x , and y, x, a s a, y, x . Part Ž.iii follows from Žb, x, a .sy Ža, x, b ., while Ž. iv is an easy consequence wxwx of partsŽ. i , Ž ii . , and Ž iii . . Now, by Ž 2 . we have that Ž Lab, L , T ab, T , wxwxwxwxy y wxwxwx Lab, L .Ž, L ab, R , T ab, R , L ab, T .Ž, and R ab, R , R ab, R , T ab, T . lie in TderŽ.A . Adding up these elements and usingŽ. ii , we obtain a ternary derivation that represents the derivation Da, b. FromŽ. iv we get s y wx Da,b adw a, bx 3 Lab, R . Finally, q wxs q wxwxwxq y adw a, bx adab , ad adw a, bx Lab, L R ab, R 2 L ab, R s y wxs 2adŽ.w a, bx 3 Lab, R 2 D a, b .

As we saw in the case of tensor products of octonions, NaltŽ.A may not be a subalgebra of A. The natural product on NaltŽ.A seems to be the commutator wxa, b s ab y ba. ¨ g wxg ¨ PROPOSITION 4.3. Gi en a, b NaltŽ.A then a, b NaltŽ.A . Moreo er, wx ¨ ŽŽ.Nalt A ,,. is a Malce algebra. s wxwxq s Proof. By Lemma 4.2Ž. iv , Lw a, bx Lab, L 2 R ab, L and Rw a, bx ywxwxy R ab, R 2 L ab, R , thus byŽ. 2 we obtain wxwq y xwxwq y xg Ž.Lw a, bx , Tab, T 2 R ab, T , L a, L b2 T a, L b TderŽ.A . Since s wxwxwxwxy s q y Tw a, bx La, L bR a, R bT ab, T 2 R ab, T and y sywxwxwxwxy s q y Lw a, bx Lab, L 2 R ab, L L ab, L 2 T a, L b, y g it follows that Ž.Lw a, bxw, T a, bxw, L a, bxwTderŽA.Ž. Similarly, R a, bx, y g wxg Rw a, bxw, T a, bx.Ž.Tder A . Therefore, a, b NaltŽ.A . The same argu- wx wx ments as those in 27, p. 9 show thatŽŽ. Nalt A , ,. is a Malcev algebra. ON THE TENSOR PRODUCT OF COMPOSITION ALGEBRAS 59

In the following we will always assume that A is simple, finite dimen- sional and generated by NaltŽ.A . In our discussion, the Lie algebra TA Ž. g generated by Ä Laa, R : a NaltŽ.A 4 will play a prominent role. Given a subset S of an algebra we say that ␦Ž.x is the degree of x on s g S if x can be written as x psŽ.1,...,sm with s1,...,sn S and pxŽ.1,..., xn some nonassociative polynomialŽ constants are allowed. of degree ␦Ž.x , and if there is no other such expression for a polynomial of degree - ␦Ž.x . By convention the degree of 0 is set to yϱ. : LEMMA 4.4. If S NaltŽ.A then the degree of x on S is the same as the g degree of Lxx, Ronspan Faa²:L , R : a S .

Proof. We proceed by induction to see that the degree of Lxxand R is F ␦Ž.x . The case ␦Ž.x syϱ is trivial. If ␦Ž.x s 0 then 0 / x g F and ␦ s s ␦ ) therefore Ž.Lxx0 Ž.R . Now let x be a monomial of degree n 1, s ␦ - ␦ ␦ - ␦ so x xx12with Ž.xix Ž.x . By induction ŽL . Ž.x , and therefore s s g ␦ - ␦ 1 x ax00or x xawith a S and Ž.x0 Ž.x . By Lemma 4.2Ž. i and ii ␦ ␦ F ␦ and by the hypothesis of induction we get Ž.Ž.Lxx, R Ž.x . Finally, s s ␦ F ␦ ␦ since x LxxŽ.1 R Ž.1 , it follows that Ž.Ž.Ž.x Lxx, R .

PROPOSITION 4.5. A is an irreducible TŽ.A - and TŽ.A s TЈ Ž.A [ F id, with TЈŽ.A s wTŽ.A ,T Ž.Ax a semisimple Lie algebra. Proof. By Lemma 4.4, the multiplication algebra of A is generated by g the left and right multiplication maps Laa, R for a NaltŽ.A . Therefore, any TŽ.A -submodule is an ideal, hence A irreducible. Since A is irre- ducible and faithful, TЈŽ.A is semisimple and T Ž.A is the direct sum of TЈŽ.A and the centerwx 19, p. 47 . But any element in the center commutes with the multiplication algebra of A and therefore lives in the centroid of A. Since A is simple and F is algebraically closed, we conclude that the center is F id. Recall that a Malcev algebra is semisimple if 0 is the only Abelian ideal wx24 . X ¨ PROPOSITION 4.6. NaltŽ.A is a semisimple Malce algebra. X s Proof. Let I be an ideal of NaltŽ.A and consider TI g g X : Ј span Faaa² L , R , D , c : a I, c NaltŽ.:A . By Lemma 4.2Ž. iv , TI T Ž.A . wxwxs g Moreover, PartŽ. v of the same lemma shows that Lab, R R ab, L T I g g X wx if a I and b NaltŽ.A . Then, by PartŽ. iv , it follows that Lab, L , wxR , R g T , too. Finally, DbŽ.g I by Ž. v , so wxD , L s L , ab I a, ca, cb Da, cŽ b. wxD , R s R g T . Therefore T is an ideal of TЈŽ.A . By the a,cb Da, cŽb. II semisimplicity of TЈŽ.A we must have wxT , T s T . In particular, I s II I X T Ž.1 s wxT , T Ž.1 s wxI, I and therefore the only abelian ideal of N Ž.A IIIX alt is 0, so NaltŽ.A is semisimple. 60 MORANDI, PEREZ´¨-IZQUIERDO, AND PUMPLUN

In a Malcev algebra M the subspace generated by the Jacobians is an ideal of M denoted by JMŽ., M, M . The subspace NM Ž.s Ä x g M : JxŽ., M, M s 04 is also an ideal and is called the J-nucleus of M.Itis well-known that NMJMŽ.Ž, M, M .s 0. In fact, any finite dimensional semisimple Malcev algebra M over a perfect field of characteristic not two can be decomposed as M s NMŽ.[ JM Ž, M, M .with NM Ž.a semisim- ple Lie algebra and JMŽ., M, M the direct sum of simple non-Lie Malcev algebras. If the field has characteristic 0 then NMŽ.is the direct sum of simple Lie algebras, bywx 16, Theorem 5.3 . X Ž.s [ XXŽ. PROPOSITION 4.7. Let Nalt A i Ni be the decomposition of Nalt A ¨ s as the direct sum of ideals that are simple Malce algebras, and let Ai X s q alg² Nii,1:Ž .Then, A is a simple unital algebra generated by Nalt Ai.F1 X ( m Nii, and A i A . g XXg / Proof. Given a Nijand b N with i j, then by Lemma 4.2Ž. v we s wx s have Da, baba3 L , R and D , bŽŽ..Nalt A 0. Since A is generated by wxs NaltŽ.A , it follows that Lab, R 0. By Lemma 4.2Ž. iv , we also get wxwxs s Lab, L R ab, R 0. By Lemma 4.4, the left and right multiplication operators by elements of Aijcommute with those by elements of A . ␸ m ª Therefore, we have an epimorphism : i Ai A given by the multipli- cation of the factors. Consider the ideals T X of TЈŽ.A as in the proof of Ni Proposition 4.6. Since TЈŽ.A is semisimple so is T X . The subalgebra A is Nii a T X-module, and by Weyl’s theorem it is completely reducible. In fact, by Ni Lemma 4.4 any submodule is an ideal and the converse. Thus A is the X i direct sum of simpleŽ. unital ideals. Fix Ai to be one of these simple X : ideals. Clearly Ł iiA A is a TAŽ.-submodule. By irreducibility it follows s X Y that A Ł jjA . Any other simple ideal Aiin the decomposition of Ai YYs s X XYs s X verifies AiijAA ŽŁ / ijAAA. ii 0. So, AiA iis a central simple m ␸ algebra as well as i Ai, and consequently is an isomorphism. Finally, : : q X s we observe that Ýi NaltŽ.Ai Nalt Ž.A F1 ÝiiN implies NaltŽ.Ai q X F1 Ni . X This proposition allows us to distinguish two cases, algebras in which N X alt is a simple Lie algebra and algebras in which Nalt is a simple non-Lie Malcev algebra. X PROPOSITION 4.8. If NaltŽ.A is a simple Lie algebra, then A is a simple associati¨e algebra. s g wx Proof. Since JaŽ.Ž., c, b 6 a, c, b for any a, b, c NaltŽ.A 27, 37 , the hypothesis implies that Ž.a, b, c s 0 and thus DcŽ.s adw x Ž.c s X a, b a, b wwa, b x, c x by Lemma 4.2. Since N Ž.A is a simple Lie algebra, any X alt derivation of N Ž.A has the form adw x ; thus, we obtain an epimorphism alt a, b X from the derivations of A onto the derivations of NaltŽ.A that is in fact an g X isomorphism because A is generated by NaltŽ.A . Given a NaltŽ.A ,we ON THE TENSOR PRODUCT OF COMPOSITION ALGEBRAS 61

X denote by Daathe unique derivation of A that restricts to ad over NaltŽ.A . y g X It is not difficult to check that span Fa² D ad a : a NaltŽ.:A is an ideal Ј Ј of T Ž.A that kills NaltŽ.A . Since T Ž.A is semisimple then the subspace killed by an ideal is a submodule of A and, by irreducibility, it must be all s y q y y of A. Therefore, ad aaD is a derivation. But Ž LaaaaaR , T R , L y y y g y g Ta.Ž, L aR aa, L R aa, L R a.Ž.ŽTder R implies 0, 3R a, 3L a. TderŽ.A which can be written as Ž.x, a, y s 0. Thus Ž.Ž.a, x, y s x, y, a s y s g X Ž.x, a, y 0 and a N Ž.A . Since A is generated by NaltŽ.A and N Ž.A is a subalgebra, this finishes the proof. X Now we will assume that NaltŽ.A is a simple non-Lie Malcev algebra, X s Ј s that is, NaltŽ.A C where C ZorŽ.F denotes the split octonion alge- bra, which is the only octonion algebra up to isomorphism over an algebraically closed field.

PROPOSITION 4.9. We ha¨e that s g X Ž.i Der ŽA .span Fa ²D , b : a, b NaltŽ.:A is a simple Lie algebra of type G2 . g X Ž.ii span Fa ²D , ba,ad:a, b NaltŽ.:A is a simple Lie algebra of type B3. Ј Ž.iii T Ž.A is a simple Lie algebra of type D4. ␨ ␩ Ј ª Ј ¨ ␨ ¬ ¬ y Ž.iv The maps , : T Ž.A T Ž.Agien by : LaaaT , R R a, ¬ ␩ ¬ y ¬ ¬ Da, baD , baaaaa, and : L L , R T , D , baD, b can be identified with the automorphisms corresponding to the permutations Ž.13 and Ž.14 of

the Dynkin diagram of D4. X Proof. Any derivation of A induces a derivation of N Ž.A , and since alt X A is generated by N Ž.A then any two derivations that agree on N Ž.A alt X alt < X must be equal. It is known that DerŽ N Ž..²A s D N Ž A. : a, b g X alt a, b alt NaltŽ.:A ; therefore, this yields the first part ofŽ. i . Consider a standard ¨ ¨ ¨ wx s y basis Ä4e12, e , u 1, u 2, u 312, , , 3of C 11 and f e1e 2. Relative to the subalgebra H s span ²:D ¨ , D ¨ , L , R , TЈŽA.decomposes as the Fu11, u 22, ff direct sum of root spaces in the way given in Table IŽ note that the elements in the right column are not 0 by evaluating them in appropriate elements of the standard basis. . Therefore, H is a Cartan subalgebra of TЈŽ.A and the root system ␨ corresponds to a simple Lie algebra of type D4. The automorphism ␣ leaves H invariant and permutes the root spaces corresponding to 1 and ␣ ␣ ␣ ␨ 32, but fixes those corresponding to and 4. Thus, can be thought of ␩ as the automorphismŽ. 13 of the Dynkin diagram of D4. Similarly, corresponds toŽ. 14 . g X The subalgebra span Fa² D , ba,ad :a, b NaltŽ.:A is the algebra fixed by the automorphism ␨␩␨ which corresponds to the automorphismŽ. 34 of the Dynkin diagram. This algebra is known to be a simple Lie algebra of type 62 MORANDI, PEREZ´¨-IZQUIERDO, AND PUMPLUN

TABLE I

Root Element that spans the root space

␣ y y 1 LuueR D , u ␣ 3313 2 Du , ¨ ␣ 23q y 3 Luue2 R D , u ␣ 3313q q 4 2 LuueR D , u ␣ q ␣ y3313y 12 LuueR D , u ␣ q ␣ 2212q y 23 Luue2 R D , u ␣ q ␣ 2212q q 24 2 LuueR D , u ␣ q ␣ q ␣ 2212q q 123 2 L¨¨R De , ¨ ␣ q ␣ q ␣ q1121y 124 L¨¨2 R De , ¨ ␣ q ␣ q ␣ 1121y y 234 L¨¨R De , ¨ ␣ q ␣ q ␣ q ␣ 1121 1234 D¨ , u ␣ q ␣ q ␣ q ␣ 13 12342 D¨ , u 12 ␣ y␣ Negative roots Change iiby , e12by e and ¨ uiiby in the previous rows

s g X B3. Finally, DerŽ.A span Fa ²D , b : a, b NaltŽ.:A is the algebra fixed by the automorphism ␩␨ , which corresponds toŽ. 134 as an automorphism of

the Dynkin diagram, therefore it is a Lie algebra of type G2 . ␣ ␣ Let Ä414,..., be a basis of a root system of D4as in the above table ␭ ␭ ␭ and 14,..., be the corresponding fundamental weights. If is a dominant weight, we will denote by VŽ.␭ the irreducible module of highest weight ␭.

PROPOSITION 4.10. The Lie algebra A is isomorphic as a D4-module to ␭ VnŽ.1 for some n.

Proof. Since F1 is a trivial submodule for B3, the branching rules for : wx the inclusion B34D 14, Theorem 8.1.4 imply that there exists an n such ( ␭ that A VnŽ.1 . ( ␭ Since C VŽ.1 , this proposition allows us to identify A with the m иии m ¨ m иии m ¨ ¨ submodule of C FFC generated by 000with the highest weight of C. This submodule obviously lies in SymnŽ.C and it is killed by the contraction SymnŽ.C ª Symny2 Ž.C , x m иии m x ¬ nxxŽ. m иии m x. In fact, this is the kernel of this contractionwx 13, Example 19.21 . Finally, in order to prove Theorem 4.1 we have to determine the product ␭ m ␭ ª on A. This product is not a D41- of VnŽ.F Vn Ž.1 ␭ VnŽ.14since that would imply that D acts as derivations, which is not ␰ true. Given an automorphism of D4 and V a module, we denote by V␰ the vector space V but with a new action given by d( x s ␰ Ž.dxfor all ON THE TENSOR PRODUCT OF COMPOSITION ALGEBRAS 63

g g d D4 and x V. ThenŽ. 2 implies that ␭ m ␭ ª ␭ VnŽ.1 ␨ FVn Ž.11␩ Vn Ž. x m y ¬ xy ␭ ( ␭ ␭ ( ␭ is a D41-homomorphism. Since VnŽ.␩ Vn Ž.31and Vn Ž.␩ Vn Ž.4 , ␭ m ␭ ␭ the product is a D43-homomorphism from VnŽ.F Vn Ž.41onto Vn Ž.. However, since

dim Hom Vn␭ m Vn␭ , Vn␭ s 1 Ž.D4Ž.Ž.3 F Ž.Ž.41 wx25 , then we only have a possibility that is fulfilled by the induced product of SymnŽ.C . This proves Theorem 4.1.

Remark 4.11. The commutative nucleus KTŽŽ..na C is killed by ad for any a, so it is killed by the action of B3. Since the decomposition of ␭ s VnŽ.13as a B -module is multiplicity free this implies that KTŽŽ..n C F. The generalized alternative nucleus NaltŽŽ..TCn is the direct sum of simple Malcev ideals. One of these ideals is F and the other is CЈ. Since any other ideal would be killed by the action of B3 we have that s s NaltŽŽ..TCnnC. In particular, NTŽŽ.. C F. s m иии m If A is as in Theorem 4.1 and A A1 FFmA is the decomposi- tion as a tensor product given by the theorem, then Proposition 3.3 implies that s m m иии mqиии q m иии m m Nalt Ž.A Nalt ŽA 1 .FFF FF F FFF NaltŽ.Am .

As in Corollary 3.10, this implies that the decomposition is unique up to order and isomorphism of the factors.

5. CONNECTIONS WITH STRUCTURABLE ALGEBRAS

Inwx 6 Allison classified the finite dimensional central simple structurable algebras over fields of characteristic zero. Later, Smirnovwx 33, 34 showed that there was a gap in the list provided by Allison, and one has to include in the previous list the algebra of symmetric octonion tensorswx 7 , a

35-dimensional algebra which in our notation corresponds to TC2Ž.. We want to analyze the connection between structurable algebras and algebras generated by its generalized alternative nucleus. Recall fromwx 6 that any structurable algebra Ž A,y. is skew-alternative; that is, the skew- symmetric elements for the involution lie in the generalized alternative nucleus. In his work, Allison first reduces the classification of finite dimensional central simpleŽ. as algebras with involution structurable alge- 64 MORANDI, PEREZ´¨-IZQUIERDO, AND PUMPLUN bras to the case in which the algebra is central simple, so up to a scalar extension one may assume that the field is algebraically closed. After that, he splits the proof into two cases, depending on whether or not the algebra is generated by the skew-symmetric elements. If the algebra is not gener- ated by the skew-symmetric elementsŽ. Case 1 , then one obtains either a central simple Jordan algebra with the identity as involution, an algebra constructed from a nondegenerate Hermitian form on a module over a unital central simple associative algebra with involution, or an algebra with involution constructed from an admissible triple. The second caseŽ. Case 2 , where the Kantor᎐Smirnov structurable algebra is missed, deals with algebras generated by the skew-symmetric elements. In this case Allison and Smirnov obtain that the only possibilities are either a central simple associative algebra with involution, the tensor product of an octonion algebra with a composition algebra, or the Kantor᎐Smirnov structurable algebra. Since the skew-symmetric elements lie in the generalized alternative nucleus, then Case 2 falls naturally into our context. So we can use Theorem 4.1 to give a new proof of this case. We will assume that A is a finite dimensional central simple structurable algebra, over an alge- braically closed field of characteristic zero, which is generated by the skew-symmetric elements. The key point is Lemma 14 in Allison’s paper which establishes that A is spanned by Ä s, s2 : s is skew-symmetric4 . Let us s m иии m write A A1 FFmA as given by Theorem 4.1. Since for any skew- symmetric element s,

2 g m иии m m m m иии s, s Ý F FFFiFFF A F i, j m m m m иии m FFjFFF A F FF, F s then m 2. If m 1, then A is either associative or isomorphic to TCnŽ. G ( with n 2. Since NaltŽŽ..TCn C then, in the latter case, Lemma 14 also s ␭ F F implies that dim TCnŽ.dim Vn Ž1 . 35, so n 2, and we obtain the octonions and the Kantor᎐Smirnov structurable algebra. Finally, if m s 2 s m s m q m then A A1 F A21and Lemma 14 implies that A A FFF F A2 q m NaltŽ.A 1 F Nalt Ž.A 2. In particular, A 1and A 2 are alternative, so A is either the tensor product of two octonion algebras or the tensor product of an octonion algebra and an associative algebra. In the second case, the involution of A preserves the associative nucleus and its centralizer so it preserves each factor in the tensor product. If Si denotes the skew-sym- s m q m metric elements of Ai, then Lemma 14 implies that A A1 FFF F q m A21S F S2 , so the set of symmetric elements of Aiimust be F and A are quadratic algebras. Therefore the associative factor must be isomor- phic to the two-by-two matrices, which is isomorphic to the . ON THE TENSOR PRODUCT OF COMPOSITION ALGEBRAS 65

6. INVARIANT BILINEAR FORMS

A symmetric bilinear formŽ. , : A = A ª F on an F-algebra A is said to be associati¨e if Ž.Ž.xy, z s x, yz for any x, y, z g A. If the algebra has an involution x ¬ x with Ž.Ž.x, y s x, y , the new bilinear form ²:x, y s Ž.x, y is symmetric and verifies ²:²:x, y s x, y and ²xy, z :²s y, xz :; that is,²: , is invariant. Inwx 33 , Schafer proves that, up to scalar multiples, there is only one invariant symmetric bilinear form on a finite dimensional central simple structurable of characteris- tic zero. That invariant form was constructed by Allison inwx 6 . In this section we construct an associative symmetric bilinear form on any algebra generated by its generalized alternative nucleus.

PROPOSITION 6.1. Let A be an algebra generated by its generalized alterna- ti¨e nucleus; then the symmetric bilinear form s Ž.x, y traceŽ.LLxy ¨ s is associati e. If A is unital, thenŽ. x, y trace Ž.Lxy . Proof. We prove that Ž.Ž.xy, z s y, zx by induction on the degree of x s g on NaltŽ.A .If x a Nalt Ž.A then s q traceŽ.LLay z traceŽ.LLLa y zR a, LL y z s q traceŽ.LLLyzaR a, LL y z s y wxq traceŽ.LLyzaLL y z, R aR a, LL y z s q traceŽ.LLyzaR a, LL yz s traceŽ.LLyza, where we have used Lemma 4.2. Thus Ž.Ž.ay, z s y, za and we get the first s s g step in the induction. Now suppose that x ax00or x xawith a s NaltŽ.A and that Žxy0 , z . Žy, zx0 .for any y, z. In the first case it follows that s s q Ž.Ž.xy, z Ž.Ž.Ž.ax00 y, z axy Ž., z Ža, x 0, yz . s q Ž.Ž.y, Ž.za x00 Ža, x , yz . s q q Ž.y, zx Ž.Ž.y, Žz, a, x00 . Ža, x , yz . s y q Ž.y, zx Ž.Ž.y, Ža, z, x00 . Žx , y, az . s y, zx y y, R , Lzq R , Ly, z Ž.Ž.xa00Ž.Ž.ax Ž. s y, zx y R , Ly, z q R , Ly, z s y, zx . Ž.Ž.ax00 Ž. Ž.axŽ.Ž. 66 MORANDI, PEREZ´¨-IZQUIERDO, AND PUMPLUN

The second case is analogous. This completes the induction. If A is unital, s s then Ž.Ž.x, y xy,1 trace Ž.Lxy . We denote the radical of this bilinear form by Rad. Since the form is associative, Rad is an ideal. We remark that if this form is nondegenerate then it is, up to scalar multiples, the only nondegenerate associative bilinear form on A wx9.

COROLLARY 6.2. If Ž., is nondegenerate then A is the direct sum of algebras as in Theorem 4.1.

Proof. Let I be an ideal with I 2 s 0 and x g I. Given y g A, : s s s xyxyAŽŽŽ ...xI 0; thus LLxyis nilpotent and Žx, y . 0. Since Rad s wx 0, we obtain I 0. By 31, Theorem 2.6 , A is the direct sum of ideals Ai Ž.s Ž[ .s that are simple unital algebras. Moreover, Nalt A Nalt i Ai [ Ž. s ² Ž.: i Nalt Aiiimplies that A alg Nalt Ai .

Remark 6.3. Let A be generated by NaltŽ.A and suppose that the s associative bilinear form Ž.x, y trace ŽLLxy .is nondegenerate. If A is unital, the corollary yields the classification of A. In general, we consider the unital algebra A࠻ s A [ F1, which contains A as an ideal. Since the bilinear forms on A and A࠻ agree, it follows that A࠻ s A [ Fe with Fe the orthogonal complement of A, which is an ideal. Now e s ␣1 q x with x g A, and ␣ / 0 implies that 0 / e2 g Fe, and we can assume that e is s ࠻ an idempotent. Therefore traceŽ.LLee 1 and the bilinear form on A is nondegenerate. By the corollary, A࠻ is the direct sum of simple unital ideals, but A is an ideal and thus it is the sum of some of these ideals. This implies that A must be unital if the bilinear form is nondegenerate.

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