The Allison-Faulkner construction of E_8
aa r X i v : . [ m a t h . R A ] F e b THE ALLISON–FAULKNER CONSTRUCTION OF E VICTOR PETROV AND SIMON W. RIGBY
Abstract.
We show that the Tits index E cannot be obtained by means of the Titsconstruction over a field with no odd degree extensions. We construct two cohomologicalinvariants, in degrees 6 and 8, of the Tits construction and the more symmetric Allison–Faulkner construction of Lie algebras of type E and show that these invariants can be usedto detect the isotropy rank. Introduction
Jacques Tits in [20] proposed a general construction of exceptional Lie algebras over anarbitrary field of characteristic not 2 or 3, now called the Tits construction. The inputs arean alternative algebra and a Jordan algebra, and the result is a simple Lie algebra of type F , E , E or E , depending on the dimensions of the algebras. The construction produces,say, all real forms of the exceptional Lie algebras, and a natural question is if all Tits indicescan be obtained this way. Skip Garibaldi and Holger Petersson in [9] showed that it is notthe case for type E , namely that Lie algebras of Tits index E do not appear as a resultof the Tits construction. We show a similar result for type E , namely that Lie algebras ofTits index E cannot be obtained by means of the Tits construction, provided that thebase field has no odd degree extensions. The proof uses the theory of symmetric spaces andthe first author’s result with Nikita Semenov and Skip Garibaldi about isotropy of groups oftype E in terms of the Rost invariant [10].We prefer to use a more symmetric version of Tits construction due to Bruce Allison andJohn Faulkner. Here the input is a so-called structurable algebra with an involution (say, thetensor product of two octonion algebras) and three constants. The Lie algebra is given bysome Chevalley-like relations. The Tits construction and the Allison–Faulkner constructionhave a large overlap but, strictly speaking, neither one is more general than the other. TheTits construction is capable of producing Lie algebras of type E whose Rost invariant hasa non-zero 3-torsion part (necessarily using a Jordan division algebra as input), but theAllison–Faulkner construction of E cannot do this – at least when the input is a form of thetensor product of two octonion algebras, because these can always be split by a 2-extension ofthe base field. On the other hand, the Allison–Faulkner construction is capable of producingLie algebras of type E with the property that the 2-torsion part of their Rost invarianthas symbol length 3, and this is impossible for the Tits construction (see [6, 11.6]). An E with this property would necessarily come from what we call an indecomposable bi-octonionalgebra, and these are related to some unusual examples of 14-dimensional quadratic formsdiscovered by Oleg Izhboldin and Nikita Karpenko [11]. Date : February 9, 2021.The first author was supported by RFBR grant 19-01-00513. The second author was supported by FWOproject G004018N.
We produce two new cohomological invariants, one in degree 6 and one in degree 8, andshow that these invariants can be used to detect the isotropy rank of either the Tits orthe Allison–Faulkner construction (but unlike the results of [9] we give necessary conditionsonly). The main tool for constructing these invariants is a calculation of the Killing form ofan Allison–Faulkner construction which, under a mild condition on the base field, is near toan 8-Pfister form (a so-called Pfister neighbour).2.
Preliminaries
Let K be a field of characteristic not 2 or 3. If q is a quadratic form, we write q ( x, y ) = q ( x + y ) − q ( x ) − q ( y ). If A is an algebra and a ∈ A , we denote by L a , R a ∈ End( A ) the left-and right-multiplication operators, respectively. Bi-octonion algebras. A K -algebra with involution ( A, − ) is called a decomposable bi-octonion algebra if it has two octonion subalgebras C and C that are stabilised by theinvolution, such that A = C ⊗ K C . A bi-octonion algebra is an algebra with involution( A, − ) that becomes isomorphic to a decomposable bi-octonion algebra over some field ex-tension. These are important examples of central simple structurable algebras, as defined byAllison in [1], and they are instrumental in constructing Lie algebras of type E (see 2.8).Any bi-octonion algebra ( A, − ) is either decomposable or it decomposes over a unique qua-dratic field extension E/K . In the latter case, there exists an octonion algebra C over E ,unique up to k -isomorphism, from which ( A, − ) can be reconstructed as follows. Let ι bethe non-identity automorphism of E/K , and let ι C be a copy of C as a k -algebra, but witha different E -algebra structure given by e · z = ι ( e ) z . Then ( A, − ) is precisely the fixed pointset of ι C ⊗ E C under the k -automorphism x ⊗ y y ⊗ x , with the involution being therestriction of the tensor product of the canonical involutions on ι C and C [2, Theorem 2.1].We denote this algebra by ( A, − ) = N E/K ( C ).To unify the description of both decomposable and non-decomposable bi-octonion algebras,if we consider C = C × C as an octonion algebra over the split quadratic ´etale extension K × K , then N K × K/K ( C ) as defined above is just isomorphic to C ⊗ K C . Multiplicative transfer of quadratic forms.
Markus Rost defined a multiplicative ana-logue of the Scharlau transfer for quadratic forms, and it has been studied by him and hisstudents (e.g., in [15, 22]) and used before to define cohomological invariants.If ( q, V ) is a quadratic space over a quadratic ´etale extension
E/K , one defines the quadraticspace ( ι q, ι V ) where ι is the non-trivial automorphism of E/K , ι V is a copy of V as a k -vectorspace but with the action of E modified by ι , and ι q ( v ) = ι ( q ( v )). Then the multiplicativetransfer N E/K ( q ) of q is the quadratic form obtained by restricting ι q ⊗ E q to the k -subspaceof tensors in ι V ⊗ E V fixed by the switch map x ⊗ y y ⊗ x .In the case of a split extension, a quadratic form over K × K is just a pair ( q , q ) where q , q are quadratic forms over K of the same dimension, and N K × K/K ( q , q ) = q ⊗ q . Let ( A, − ) = N E/K ( C ) for an octonion algebra C over a quadratic ´etaleextension E/K , and let n be the norm of C . Then N E/K ( n ) equals the normalised traceform ( x, y ) tr( L x ¯ y + y ¯ x ) . HE ALLISON–FAULKNER CONSTRUCTION OF E Proof.
Both N E/K ( n ) and the normalised trace form are invariant symmetric bilinear formson ( A, − ) in the sense that Allison defined (see [1, Theorem 17] and [2, Proposition 2.2]).By a theorem of Schafer [16], a central simple structurable algebra has at most one suchbilinear form, up to a scalar multiple. (As discussed in [16, p. 116–117], these facts arevalid in characteristic 0 or p ≥
5, despite some of the original references being limited tocharacteristic 0.) (cid:3)
Lie-related triples.
Let ( A, − ) be a central simple structurable algebra over K . A Lierelated triple (in the sense of [4, § T = ( T , T , T ) where T i ∈ End( A ) and T i (cid:0) xy (cid:1) = T j ( x ) y + xT k ( t )for all x, y ∈ A and all ( i j k ) that are cyclic permutations of (1 2 3). Define T to be theLie subalgebra of gl ( A ) × gl ( A ) × gl ( A ) spanned by the set of related triples.For a, b ∈ A and 1 ≤ i ≤
3, define T ia,b = ( T , T , T )where (taking indices mod 3): T i = L ¯ b L a − L ¯ a L b ,T i +1 = R ¯ b R a − R ¯ a R b ,T i +2 = R ¯ ab − ¯ ba + L b L ¯ a − L a L ¯ b . Let T I be the subspace of End( A ) spanned by { T ia,b | a, b ∈ A, ≤ i ≤ } . Since ( A, − ) isstructurable, T I is a Lie subalgebra of T [4, Lemma 5.4]. Finally, denote by Skew( A, − ) ⊂ A the ( − T ′ be the subspace of End( A ) spanned bytriples of the form(2.4.1) ( D, D, D ) + ( L s − R s , L s − R s , L s − R s )where D ∈ Der( A, − ) and s i ∈ Skew( A, − ) with s + s + s = 0. Let ( C, − ) be an octonion algebra with norm n . The principle of local trialityholds in T I in the sense that each of the projections T I → gl ( C ), ( T , T , T ) T i , for1 ≤ i ≤
3, is injective [19, Theorem 3.5.5]. The Lie algebra T I is isomorphic to so ( n ) [19,Lemma 3.5.2]. The ( i + 2)-th entry of the triple T ia,b is R ¯ ab − ¯ ba + L b L ¯ a − L a L ¯ b , and by [19,pp. 51, 54] this is the map C → C that sends(2.5.1) x n ( x, a ) b − n ( x, b ) a. If ( A, − ) is a bi-octonion algebra of the form ( A, − ) = N E/K ( C ) forsome quadratic ´etale extension E/K and some octonion algebra C over E , then T I = T = T ′ ≃ Lie( R E/k ( Spin ( n ))) , where n is the norm of C .Proof. We have that T I ⊂ T ⊂ T ′ and dim T ′ = dim Der( A, − ) + 2 dim Skew( A, − ) =28 + 28 = 56 by [4, Corollary 3.5]. On the other hand, T I (as an E -module) is preciselyLie( Spin ( n )) [19, Theorem 3.5.5] and so T I (as a K -vector space) is 56-dimensional andisomorphic to Lie( R E/K ( Spin ( n ))). (cid:3) VICTOR PETROV AND SIMON W. RIGBY
Local triality.
In the context of Proposition 2.6, the Lie algebra T I is of type D + D .Local triality holds here too: the projections T I → gl ( A ) , ( T , T , T ) T i are injective forany 1 ≤ i ≤
3, and the symmetric group S acts on T I by E -automorphisms, where E is thecentroid of T I (compare with [19, § The Allison–Faulkner construction [4, § . Let ( A, − ) be a central simple structurablealgebra and let γ = ( γ , γ , γ ) ∈ K × × K × × K × . For 1 ≤ i, j ≤ i = j , define A [ ij ] = { a [ ij ] | a ∈ A } to be a copy of A , and identify A [ ij ] with A [ ji ] by setting a [ ij ] = − γ i γ − j a [ ji ].Define the vector space K ( A, − , γ ) = T I ⊕ A [12] ⊕ A [23] ⊕ A [31]and equip it with an algebra structure defined by the multiplication:[ a [ ij ] , b [ jk ]] = − [ b [ jk ] , a [ ij ]] = ab [ ik ] , [ T, a [ ij ]] = − [ a [ ij ] , T ] = T k ( a )[ ij ][ a [ ij ] , b [ ij ]] = γ i γ − j T ia,b for all a, b ∈ A , T = ( T , T , T ) ∈ T I , and ( i j k ) a cyclic permutation of (1 2 3). Then K ( A, − , γ ) is clearly a Z / Z × Z / Z -graded algebra, and it is in fact a central simple Liealgebra [4, Theorems 4.1, 4.3, 4.4, & 5.5]. Relation to the Tits–Kantor–Koecher construction.
If the quadratic form h γ , γ , γ i isisotropic then K ( A, − , γ ) ≃ K ( A, − ) where K ( A, − ) = Skew( A ) ⊕ A ⊕ V A,A ⊕ A ⊕ Skew( A )is the Tits–Kantor–Koecher construction [3, Corollary 4.7]. An isomorphism and its in-verse are determined explicitly in [3, Theorem 2.2] in the case where − γ γ − is a square.More generally, if h γ , γ , γ i and h γ ′ , γ ′ , γ ′ i are similar quadratic forms, then K ( A, − , γ ) ≃ K ( A, − , γ ′ ) [3, Proposition 4.1].In particular, if ( A, − ) is a bi-octonion algebra, then K ( A, − , γ ) is a central simple Liealgebra of type E . Relation to the Tits construction.
Jacques Tits in [20] defined the following construc-tion of Lie algebras. Let C be an alternative algebra and J be a Jordan algebra. Denoteby C ◦ and J ◦ the subspaces of elements of generic trace zero and define operations ◦ andbilinear forms ( − , − ) on C ◦ and J ◦ by the formula ab = a ◦ b + ( a, b )1 . Two elements a, b in J and C define an inner derivation h a, b i of the respective algebra,namely: h a, b i ( x ) = [[ a, b ] , x ] − [ a, b, x ] . Then there is a Lie algebra structure on the vector space Der( J ) ⊕ J ◦ ⊗ C ◦ ⊕ Der( C ) definedby the formulas[Der( J ) , Der( C )] = 0;[ B + D, a ⊗ c ] = B ( a ) ⊗ c + a ⊗ D ( c );[ a ⊗ c, a ′ ⊗ c ′ ] = ( c, c ′ ) h a, a ′ i + ( a ◦ a ′ ) ⊗ ( c ◦ c ′ ) + ( a, a ′ ) h c, c ′ i HE ALLISON–FAULKNER CONSTRUCTION OF E for all B ∈ Der( J ), D ∈ Der( C ), a, a ′ ∈ J ◦ , and c, c ′ ∈ C ◦ . If ( A, − ) = C ⊗ C is adecomposable bi-octonion algebra, then K ( A, − , γ ) is isomorphic to the Lie algebra obtainedvia the Tits construction from the composition algebra C and the reduced Albert algebra H ( C , γ ) [3, Remark 1.9 (c)]. Let ( A, − ) = C ⊗ C be a decomposable bi-octonion algebra. Then T I ⊕ A [ ij ] ≃ so ( h γ i i n ⊥ h− γ − j i n ) , where n ℓ is the norm of C ℓ .Proof. Consider the quadratic form Q = h γ i i n ⊥ h− γ − j i n on the vector space C ⊕ C .The Lie algebra so ( Q ) can be embedded into the Clifford algebra C ( Q ) as the subspacespanned by elements of the form[ u, v ] c , u, v ∈ C ⊕ C where [ − , − ] c denotes the commutator in the Clifford algebra (to avoid confusion with thecommutators in C and C ). These generators satisfy the relations [13, p. 232 (30)]:[[ u, v ] c , [ u ′ , v ′ ] c ] c = − Q ( u, u ′ )[ v, v ′ ] c + 2 Q ( u, v ′ )[ v, u ′ ] c + 2 Q ( v, u ′ )[ u, v ′ ] c − Q ( v, v ′ )[ u, u ′ ] c . (2.11.1)If z, z ′ ∈ C and w, w ′ ∈ C , this becomes[[ z, w ] c , [ z ′ , w ′ ] c ] c = − γ i n ( z, z ′ )[ w, w ′ ] c + 2 γ − j n ( w, w ′ )[ z, z ′ ] c . (2.11.2)This implies that the 64-dimensional subspace spanned by[ z, w ] c , z ∈ C , w ∈ C generates the Lie algebra so ( Q ). Now define a linear bijection θ : so ( Q ) → T I ⊕ A [ ij ] by[ z, z ′ ] c γ i T iz,z ′ , [ w, w ′ ] c
7→ − γ − j T iw,w ′ , [ z, w ] c zw [ ij ]for all z, z ′ ∈ C and w, w ′ ∈ C . By [13, p. 232 (31)] and (2.5.1), the restriction of θ to thesubalgebra [ C , C ] c ⊕ [ C , C ] c ≃ so ( h γ i i n ) × so ( h− γ − j i n ) is a homomorphism.Now we calculate using (2.11.2) that θ ([[ z, w ] c , [ z ′ , w ′ ] c ] c ) = θ ( − γ i n ( z, z ′ )[ w, w ′ ] c + 2 γ − j n ( w, w ′ )[ z, z ′ ] c )= − γ i n ( z, z ′ ) θ ([ w, w ′ ] c ) + 2 γ − j n ( w, w ′ ) θ ([ z, z ′ ] c )= 2 γ i γ − j (cid:0) n ( z, z ′ ) T iw,w ′ + n ( w, w ′ ) T iz,z ′ (cid:1) . (2.11.3)Meanwhile, we have[ θ ([ z, w ] c ) , θ ([ z ′ , w ′ ] c )] = [ zw [ ij ] , z ′ w ′ [ ij ]] = γ i γ − j T izw,z ′ w ′ (2.11.4)To complete the proof that θ is an isomorphism, we show that the triples (2.11.3) and (2.11.4)are equal. It suffices to compare the i -th entries of each triple (by § L x L x ′ + L x ′ L x = L x L x ′ + L x ′ L x = n i ( x, x ′ ) id VICTOR PETROV AND SIMON W. RIGBY for all x ∈ C ℓ [19, Lemma 1.3.3 (iii)], the i -th entry of (2.11.3) is2 γ i γ − j (cid:0) n ( z, z ′ )( L w ′ L w − L w L w ′ ) + n ( w, w ′ )( L z ′ L z − L z L z ′ ) (cid:1) = 2 γ i γ − j (cid:0) ( L z L z ′ + L z ′ L z )( L w ′ L w − L w L w ′ ) + ( L w L w ′ + L w ′ L w )( L z ′ L z − L z L z ′ ) (cid:1) = 2 γ i γ − j (2 L z ′ L z L w ′ L w − L z L z ′ L w L w ′ ) = 4 γ i γ − j ( L z ′ L z L w ′ L w − L z L z ′ L w L w ′ )In the last line, we have used (multiple times) the fact that C and C commute and associatewith each other. Using this fact a few more times, the i -th entry of (2.11.4) is just4 γ i γ − j ( L z ′ w ′ L zw − L zw L z ′ w ′ ) = 4 γ i γ − j ( L z ′ L z L w ′ L w − L z L z ′ L w L w ′ ) . (cid:3) The Killing form of K ( A, − , γ )For any quadratic form q = h x , . . . , x n i , the Killing form of so ( q ) is(3.0.1) h − n i λ ( q )where λ ( q ) = ⊥ i 1) = 1, we have λ λ ∈ K × . Therefore, q = q E | A = h λ λ i N E/K ( n ). (cid:3) We can now calculate the Killing form of K ( A, − , γ ) in the case where ( A, − ) is a bi-octonionalgebra. If ( A, − ) = N E/K ( C ) , then the Killing form on K ( A, − , γ ) is (3.2.1) h− i (cid:0) tr E/K ( λ ( n )) ⊥ h γ γ − , γ γ − , γ γ − i N E/K ( n ) (cid:1) . Proof. Let κ be the Killing form of K ( A, − , γ ). If x, y ∈ K ( A, − , γ ) are from differenthomogeneous components in the Z / Z × Z / Z -grading, then ad x ad y shifts the grading andconsequently κ ( x, y ) = tr(ad x ad y ) = 0.Let τ be the Killing form of T I . The Killing form of Lie( Spin ( n )) is h− i λ ( n ); see (3.0.1).Since T I ≃ Lie( R E/K ( Spin ( n )) by Proposition 2.6, we have τ = tr E/K ( h− i λ ( n )) = h− i tr E/K ( λ ( n )). There is an automorphism of K ( A, − , γ ) ⊗ K K alg that swaps the twosimple subalgebras of T I ⊗ K K alg , and this implies κ | T I is a scalar multiple of τ ; say κ | T I = h φ ih− i tr E/K ( λ ( n )) for some φ ∈ K × .Let us determine φ . The grading on K ( A, − , γ ) makes it a sum of four T I -modules. For T, S ∈ T I and a ∈ A , [ T, [ S, a [ ij ]]] = T k ( S k ( a ))[ ij ] . HE ALLISON–FAULKNER CONSTRUCTION OF E Therefore κ ( T, S ) = tr(ad T ad S ) = τ ( T, S ) + tr( T S ) + tr( T S ) + tr( T S ) . The trace forms of the irreducible representations T I → gl ( A ), T T ℓ for 1 ≤ ℓ ≤ T S ) = tr( T S ) =tr( T S ) for all T, S ∈ T I . Moreover, tr( T S ) is a scalar (in fact, integer) multiple of τ ( T, S ).To determine the ratio between tr( T S ) and τ ( T, S ), we can assume A = C ⊗ C is decom-posable, and consider the subalgebra so ( n ) ⊂ so ( n ) × so ( n ) ≃ Lie( R E/K ( Spin ( n )), where n ℓ is the norm on C ℓ . It is well-known that the Killing form κ on so ( n ) is 6 (= 8 − 2) timesthe trace form of its vector representation so ( n ) → gl ( C ), while the trace form of the repre-sentation so ( n ) → gl ( C ⊗ C ) is clearly 8 times the trace form of the vector representation.But κ is equal to the restriction of the Killing form τ on so ( n ) × so ( n ), so this means that(if T ∈ T I belongs to the so ( n ) subalgebra) we have tr( T ) = 8 tr( T | C ) = κ ( T ) = τ ( T ).In conclusion, φ = 5, so κ | T I = h− i tr E/K ( λ ( n )).The restriction κ | A [ ij ] is an invariant form under the action of R E/K ( Spin ( n )), which meansit is proportional to N E/K ( n ), by Lemma 3.1. Say κ | A [ ij ] = h φ ij i N E/K ( n ). To determinethe φ ij , it suffices to calculate κ (1[ ij ]), since κ (1[ ij ]) = φ ij N E/K ( n )(1) = φ ij . By definition κ (1[ ij ]) is the trace of ad ij ]2 . The graded components of K ( A, − , γ ) are invariant underad ij ]2 , so we work out the trace separately for each of these components.For all b ∈ A , we have[1[ ij ] , [1[ ij ] , b [ jk ]]] = [1[ ij ] , b [ ik ]] = − γ i γ − j [1[ ji ] , b [ ik ]] = − γ i γ − j b [ jk ]so ad ij ]2 | A [ jk ] = − γ i γ − j id, and tr(ad ij ]2 | A [ jk ] ) = − γ i γ − j . Similarly, for all b ∈ A ,[1[ ij ] , [1[ ij ] , b [ ki ]]] = ( − γ i γ − j )( − γ k γ − i )[1[ ij ] , [1[ ji ] , ¯ b [ ik ]]] = ( − γ i γ − j )( − γ k γ − i )[1[ ij ] , ¯ b [ jk ]]= ( − γ i γ − j )( − γ k γ − i )[¯ b [ ik ]] = ( − γ i γ − j )( − γ k γ − i )( − γ i γ − k ) b [ ki ]= − γ i γ − j b [ ki ] , so ad ij ]2 | A [ ki ] = − γ i γ − j id, and tr(ad ij ]2 | A [ jk ] ) = − γ i γ − j . In contrast, for all b ∈ A ,[1[ ij ] , [1[ ij ] , b [ ij ]]] = [1[ ij ] , γ i γ − j T i ,b ] = − γ i γ − j ( T i ,b ) k (1)= − γ i γ − j ( R b − ¯ b + L b − L ¯ b )(1) = − γ i γ − j ( b − ¯ b ) . Therefore, ad ij ]2 | A [ ij ] has a 50-dimensional kernel { a [ ij ] | ¯ a = a } and a 14-dimensionaleigenspace { a [ ij ] | ¯ a = − a } with eigenvalue − γ i γ − j . This proves that tr(ad ij ]2 | A [ ij ] ) = − γ i γ − j .Now if T = ( T , T , T ) ∈ T I , then[1[ ij ] , [1[ ij ] , T ]] = [1[ ij ] , − T k (1)[ ij ]] = − γ i γ − j T i ,T k (1) . We can use (2.4.1) to write T = ( D, D, D ) + ( L s − R s , L s − R s , L s − R s ) for some unique D ∈ Der( A, − ) and s i ∈ Skew( A, − ) such that s + s + s = 0. Note that the k -th entry of T is L s i − R s j . Then T k (1) = D (1) + L s i (1) − R s j (1) = s i − s j , so ad ij ]2 ( T ) = − γ i γ − j T i ,T k (1) is the triple whose k -th entry is − γ i γ − j ( R T k (1) − T k (1)1 + L T k (1) L − L L T k (1) ) = − γ i γ − j ( R s i − s j ) + L s i − s j ) ) VICTOR PETROV AND SIMON W. RIGBY = − γ i γ − j (cid:0) ( L s i − R s j ) − ( L s j − R s i ) (cid:1) . This shows ker(ad ij ]2 | T I ) is the 42-dimensional subspace of T I whose k -th projection is { D + L s − R s | D ∈ Der( A, − ) , s ∈ Skew( A, − ) } . And the subspace of T I whose k -th projection is { L s + R s | s ∈ Skew( A, − ) } is a 14-dimensional eigenspace of ad ij ]2 | T I with eigenvalue − γ i γ − j . This proves thattr(ad ij ]2 | T I ) = − γ i γ − j . Therefore φ ij = κ (1[ ij ]) = tr(ad ij ]2 ) = − γ i γ − j − γ i γ − j − γ i γ − j − γ i γ − j = − γ i γ − j , and we can simplify to get (3.2.1) because 15 is in the same square class as 240. (cid:3) If char( K ) = 5, then the form Killing form on E is zero, but deleting the factor h− i in(3.2.1) gives a nondegenerate quadratic form on K ( A, − , γ ) such that the associated bilinearform is Lie invariant (see [12, p. 117]).Assume for the rest of this section that − K ; equivalently, 4 = 0in the Witt ring W ( K ). Let ( A, − ) = N E/K ( C ) , and let κ ′ be a nondegenerate Lie invariant bilinearform on K ( A, − , γ ) . Then κ ′ ∈ I ( K ) and there is a unique -dimensional form q ∈ I ( K ) such that q + κ ′ ∈ I ( K ) .Proof. Since κ ′ is unique up to a scalar multiple, we can assume without loss of generalitythat κ ′ = (cid:0) tr E/K ( λ ( n )) ⊥ h γ γ − , γ γ − , γ γ − i N E/K ( n ) (cid:1) . Then κ ′ ∈ I ( K ), since tr E/K ( λ ( n )) = 0 [7, Lemma 19.8] and N E/K ( n ) ∈ I ( K ) [15], [22,Satz 2.16]. Setting q = N E/K ( n ) yields q + κ ′ = h , γ γ − , γ γ − , γ γ − i N E/K ( n ) = ⟪ − γ γ − , − γ γ − ⟫ N E/K ( n ) ∈ I ( K ) . The uniqueness follows from the Arason–Pfister Hauptsatz. (cid:3) Let Q ( ∗ ) ⊂ R ( ∗ ) ⊂ H ( ∗ , E ) be the functors Fields /K → Sets such that for all fields F/K :(i) Q ( F ) is the set of isomorphism classes of Lie algebras of type E that are isomorphicto K ( A, − , γ ) for some bi-octonion algebra ( A, − ) over F and some γ = ( γ , γ , γ ) ∈ ( K × ) ; i.e. Q ( F ) is the image of the Allison–Faulkner construction H ( F, ( G × G ⋊ Z / Z ) × ( Z / Z ) ) → H ( F, E ) . (ii) R ( F ) is the set of isomorphism classes of Lie algebras L of type E such that the classof L is contained in Q ( F ′ ) for some odd-degree extension F ′ /F .Recall from 2.10 that Q ( ∗ ) contains all Lie algebras of type E that are obtainable using theTits construction from a reduced Albert algebra and an octonion algebra. Whereas, R ( ∗ )strictly contains all Lie algebras of type E that are obtainable using the Tits constructionfrom an Albert algebra (even a division algebra) and an octonion algebra. Any cohomological HE ALLISON–FAULKNER CONSTRUCTION OF E invariant Q ( ∗ ) → L i ≥ H i ( ∗ , Z / Z ) can be extended uniquely to a cohomological invariant R ( ∗ ) → L i ≥ H i ( ∗ , Z / Z ) [7, § e n : I n ( ∗ ) → H n ( ∗ , Z / Z ) for n = 6 and 8, weobtain cohomological invariants of the Tits construction and the Allison–Faulkner construc-tion. If − is a sum of two squares in K , then there exist nontrivial cohomologicalinvariants h : R ( ∗ ) → H ( ∗ , Z / Z ) ,h : R ( ∗ ) → H ( ∗ , Z / Z ) such that if A = N E/F ( C ) , then h ( K ( A, − , γ )) = e ( N E/F ( n )) ,h ( K ( A, − , γ )) = ( − γ γ − ) ∪ ( − γ γ − ) ∪ e ( N E/F ( n )) . Isotropy of Tits construction In this section we continue to assume that the base field K is of characteristic not 2 or 3. E /P as a compactification of D /A ⋊ Z / Z . Consider the split group of type E over K and the action of its subgroup of type D on the projective homogeneous variety E /P . Since D being the fixed point subgroup of an involution σ is known to be spherical,the number of orbits (in the geometrical sense) must be finite, in particular, there is anopen orbit. Using [18, Lemma 2.9 and 2.11] we can give a precise description of this orbit:it consists of parabolic subgroups P of type P such that σ ( P ) is opposite to P , that is P ∩ σ ( P ) = L is a Levi subgroup of P . In particular, L is stable under the action of σ , a fortiori its commutator subgroup (of type E ) and the centralizer of the commutatorsubgroup (of type A ) are stable under the action of σ . In particular, the subgroup of type E + A generated by them is also stable under the action of σ , and the stabilizer of a pointin the open orbit is the fixed points subgroup of σ acting on E + A . Using [14, Table I] wesee that this subgroup is of type A ; more precisely, it is known to be SL /µ ⋊ Z / Z . Let [ ξ ] be in H ( K, G ) , H be a closed subgroup of G . Then ξ ( G/H ) has a K -rational point if and only if [ ξ ] comes from some [ ζ ] ∈ H ( K, H ) .Proof. See [17, Proposition 37]. (cid:3) Let [ ξ ] ∈ H ( K, PGO n ) be in the image of H ( K, GL n /µ ⋊ Z / Z ) . Thenthere exists a quadratic field extension E/K such that the orthogonal involution correspondingto ξ E is hyperbolic.Proof. Consider the following short exact sequence: H ( K, GL n /µ ) → H ( K, GL n /µ ⋊ Z / Z ) → H ( K, Z / Z ) , and take E/K corresponding to the image in H ( K, Z / Z ) of [ ζ ] in H ( K, GL n /µ ⋊Z / Z ) whose image in H ( K, PGO n ) is [ ξ ]. Passing to E we see that [ ξ E ] comes from H ( E, GL n /µ ) and so produces a hyperbolic involution. (cid:3) Let K be a -special (that is with no odd degree extensions) field of charac-teristic not and , L be a Lie algebra of type E obtained via the Tits construction. Thenthe group corresponding to L is not of Tits index E , .Proof. Assume the contrary. Obviously the base field is infinite, for there are only splitgroups of type E over finite fields. Let L be obtained via the Tits construction from C and H ( C , γ ) for some octonion algebras C and C . Denote by [ ζ ] in H ( K, E ) the classcorresponding to L . By Proposition 2.11 L contains a Lie subalgebra of type D , namely so ( h γ i i n ⊥ h− γ − j i n ), and so the corresponding group contains a subgroup of type D (see[5, Expos´e XXII, Corollaire 5.3.4]). This means that ζ ( E /D ) contains a K -rational point,and by Lemma 4.2 [ ζ ] comes from some [ ξ ] ∈ H ( K, D ).Now ξ ( E /P ) is a smooth compactification of its open subvariety U = ζ ( D /A ⋊ Z / Z )and by the assumption has a rational point. This means that there is a parabolic subgroupof type P inside ξ E , and the unipotent radical of an opposite parabolic subgroup definesan open subvariety in ξ ( E /P ) isomorphic to A . Since the base field is infinite, there is arational point in A ∩ U . Applying Lemma 4.2 and Lemma 4.3 we see that the quadraticform h γ i i n ⊥ h− γ − j i n becomes hyperbolic over a quadratic field extension E/K . It followsthat e ( n ) + e ( n ) is trivial over E , hence n − n belongs to I and so is hyperbolic over E . Now n − n is divisible by the discriminant of E and so e ( n ) + e ( n ) is a sum of twosymbols with a common slot. But the Rost invariant of the anisotropic kernel of type E is e ( n ) + e ( n ), and applying [10, Theorem 10.18] we see that this group must be isotropic,a contradiction. (cid:3) Note that [7, Appendix A] provides an example of a strongly inner group of type E over a2-special field, hence an example of a group of Tits index E , over such a field. Suppose K is a field such that − is a sum of two squares, and let L be aLie algebra over K of type E obtained via the Tits construction. (i) If h ( L ) = 0 then L is anisotropic. (ii) If − is a square in K and h ( L ) = 0 then L has K -rank ≤ .Proof. (i) Suppose L is isotropic. After an odd-degree extension F/K , we can assume that L F does not have Tits index E , by Theorem 4.4. We can also assume that L F does nothave Tits index E , because then its anisotropic kernel would be of type E , and everysuch group becomes isotropic over an odd-degree extension (see [8, Exercise 22.9]). In eachof the remaining possible indices from [21, p. 60], L F corresponds to a class in the image of H ( F, Spin ) → H ( F, E ), which means it is isomorphic to K ( A, − ) ≃ K ( A, − , (1 , − , A . Then clearly h ( L F ) = 0, so h ( L ) = 0.(ii) Suppose L has K -rank ≥ 2. Then there is an odd-degree extension F/K such that L F corresponds to a class in the image of H ( F, Spin ) → H ( F, E ). Its anisotropic kernel HE ALLISON–FAULKNER CONSTRUCTION OF E is a subgroup of Spin ( q ) for some 12-dimensional form q belonging to I ( K ), and by awell-known theorem of Pfister (see [7, Theorem 17.13]) q is similar to n − n for a pair of3-Pfister forms n i with a common slot, say n i = ⟪ x, y i , z i ⟫ . 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Classification of algebraic semisimple groups. In Algebraic Groups and Discontinuous Subgroups (1966), A. Borel and G. D. Mostow, Eds., vol. 9 of Proceedings of Symposia in Pure Mathematics , pp. 33–62.[22] Wittkop, T. Die Multiplikative Quadratische Norm F¨ur Den Grothendieck-Witt-Ring. Master’s thesis,Universit¨at Bielefeld, 2006. Petrov: St. Petersburg State University, 29B Line 14th (Vasilyevsky Island), 199178, St. Pe-tersburg, Russia; PDMI RAS, Nab. Fontanki 27, 191023, St. Petersburg, Russia Email address : [email protected] Rigby: Department of Mathematics: Algebra and Geometry, Ghent University, Krijgslaan281, 9000 Ghent, Belgium Email address ::