GROUPS OF TYPE E OVER ARBITRARY FIELDS 7 0 0 R. SKIP GARIBALDI 0 2 Abstract. Freudenthaltriple systemscomeintwoflavors,degenerateandnondegenerate. n The best criterion for distinguishing between the two which is available in the literature is a J by descent. We provide an identity which is satisfied only by nondegenerate triple systems. 9 We then use this to define algebraic structures whose automorphism groups produce all 1 adjoint algebraic groups of type E7 over an arbitrary field of characteristic=2,3. The main advantage of these new structures is that they incorporate a6 previously un- ] G considered invariant(a symplectic involution) of these groups in a fundamental way. As an application, we give a construction of adjoint groups with Tits algebras of index 2 which A provides a complete description of this involution and apply this to groups of type E7 over . h a real-closed field. t a m A useful strategy for studying simple (affine) algebraic groups over arbitrary fields has [ been to describe such a group as the group of automorphisms of some algebraic object. 3 We restrict our attention to fields of characteristic = 2,3. The idea is that these algebraic v 6 6 objects are easier to study, and their properties correspond to properties of the group one is 5 interested in. Weil described all groups of type A , B , C , 1D , and 2D in this manner in 0 n n n n n [Wei60]. Similar descriptions were soonfoundforgroupsoftypeF (asautomorphism groups 1 4 1 of Albert algebras) and G (as automorphism groups of Cayley algebras). Recently, groups 2 8 of type 3D and 6D have been described in [KMRT98, 43] as groups of automorphisms of 9 4 4 § / trialitarian central simple algebras. The remaining groups are those of types E , E , and h 6 7 E . t 8 a As an attempt to provide an algebraic structure associated to groups of type E , Freuden- m 7 thal introduced a new kind of algebraic structure in [Fre54, 4], which was later studied : v § axiomatically in [Mey68], [Bro69], and [Fer72]. These objects, called Freudenthal triple sys- i X tems, come in two flavors: degenerate and nondegenerate. The automorphism groups of the r nondegenerate ones provide all simply connected groups of type E with trivial Tits algebras a 7 over an arbitrary field. (The Tits algebras of a group G are the endomorphism rings of certain irreducible representations of G. They were introduced in [Tit66b], or see [KMRT98, 27.A] for another treatment.) In fact, more is true: they are precisely the G-torsors for G § simply connected split of type E . 7 One issue that has not been addressed adequately in the study of triple systems is how to distinguish between the two kinds. A triple system consists of a 56-dimensional vector space endowed with a nondegenerate skew-symmetric bilinear form and a quartic form (see 1.1 for a complete definition), and we say that the triple system is nondegenerate precisely when this quartic form is irreducible when we extend scalars to a separable closure of the 1991 Mathematics Subject Classification. 17A40 (11E72 14L27 17B25 20G15). 1 2 R. SKIP GARIBALDI base field. There seems to be essentially no criterion available in the literature to distinguish between the two types other then checking the definition. In Section 2, we show that one of the identities which nondegenerate triple systems are known to satisfy is not satisfied by degenerate ones, thus providing a means for differentiating between the two types which doesn’t require enlarging the base field. In Section 3, we define algebraic structures whose groups of automorphisms produce all groups of type E up to isogeny. Thanks to the preceding result distinguishing between 7 degenerate and nondegenerate triple systems, no Galois descent is needed for this definition. We call these structures gifts (short for generalized Freudenthal triple systems). They are triples (A,σ,π) such that A is a central simple F-algebra of degree 56, σ is a symplectic involution on A, and π: A A is an F-linear map satisfying certain axioms (see 3.2 for −→ a full definition). We also show an equivalence of categories between the category of gifts over an arbitrary field F and the category of adjoint (equivalently, simply connected) groups of type E over F. A description of the flag (a.k.a. homogeneous projective) varieties of an 7 arbitrary group of type E is then easily derived in Section 4. 7 The main strength of these gifts is that they include the involution σ in an intrinsic way. Specifically, as mentioned above every adjoint group G of type E is isomorphic to 7 Aut(A,σ,π) for some gift (A,σ,π). The inclusion of the split adjoint group of type E in 7 the split adjoint group of type C gives via Galois cohomology that the pair (A,σ) is an 28 invariant of the group G. We give a construction (in 5.3) of gifts with algebra component A of index 2 (which is equivalent to constructing groups of type E with Tits algebra of index 7 2 and corresponds to Tits’ construction of analogous Lie algebras in [Tit66a], or see [Jac71, 10]) and we describe the involution σ explicitly in this case. In the final section, we use this § construction to prove some facts about simple groups of type E over a real-closed field. 7 Notations and conventions. All fields that we consider will have characteristic = 2,3. 6 For g an element in a group G, we write Int(g) for the automorphism of G given by x gxg 1. − 7→ For X a variety over a field F and K any field extension of F, we write X(K) for the K-points of X. When we say that an affine algebraic group (scheme) G is simple, we mean that it is absolutely almost simple in the usual sense (i.e., G(F ) has a finite center and no noncentral s normal subgroups for F a separable closure of the ground field). s We write G for the algebraic group whose F-points are F and µ for its subgroup m,F ∗ n group of nth roots of unity. We will also follow the usual conventions for Galois cohomology and write Hi(F,G) := Hi( aℓ(F /F),G(F )) for G any algebraic group over F, and similarly for the cocycles s s G Z1(F,G). For more information about Galois cohomology, see [Ser79] and [Ser94]. We follow the notation in [Lam73] for quadratic forms. For I a right ideal in a central simple F-algebra A, we define the rank of I to be (dim I)/degA. Thus when A is split, so that we may write A = End (V) for some F ∼ F GROUPS OF TYPE E OVER ARBITRARY FIELDS 3 7 F-vector space V, I = Hom (V,U) for some subspace U of V and the rank of I is precisely F the dimension of U. 1. Background on triple systems Definition 1.1. (See, for example, [Fer72, p. 314] or [Gar99, 3.1]) A (simple) Freudenthal triple system is a 3-tuple (V,b,t) such that V is a 56-dimensional vector space, b is a nonde- generate skew-symmetric bilinear formon V, and t is a trilinear product t: V V V V. × × −→ We define a 4-linear form q(x,y,z,w) := b(x,t(y,z,w)) for x, y, z, w V, and we require ∈ that FTS1: q is symmetric, FTS2: q is not identically zero, and FTS3: t(t(x,x,x),x,y) = b(y,x)t(x,x,x)+q(y,x,x,x)x for all x,y V. ∈ We say that such a triple system is nondegenerate if the quartic form v q(v,v,v,v) on 7→ V is absolutely irreducible (i.e., irreducible over a separable closure of the base field) and degenerate otherwise. Note that since b is nondegenerate, FTS1 implies that t is symmetric. One can linearize FTS3 a little bit to get an equivalent axiom that will be of use later. Specifically, replacing x with x+λz, expanding using linearity, and taking the coefficient of λ2, one gets the equivalent formula FTS3 t(t(x,x,z),z,y)+t(t(x,z,z),x,y) = zq(x,x,z,y)+xq(x,z,z,y) ′ +b(y,z)t(x,x,z)+b(y,x)t(x,z,z). Example 1.2. (Cf. [Bro69, p. 94], [Mey68, p. 172]) Let W be a 27-dimensional F-vector space endowed with a non-degenerate skew-symmetric bilinear form s and set F W (1.3) V := . W F (cid:18) (cid:19) For α j γ k (1.4) x := and y := j β k δ ′ ′ (cid:18) (cid:19) (cid:18) (cid:19) set b(x,y) := αδ βγ +s(j,k )+s(j ,k). ′ ′ − We define the determinant map det: V F by −→ det(x) := αβ s(j,j ) ′ − and set α j t(x,x,x) := 6det(x) − . j β ′ (cid:18) − (cid:19) 4 R. SKIP GARIBALDI Then (V,b,t) is a Freudenthal triple system. Since q(x,x,x,x) := 12det(x)2, it is certainly degenerate, and we denote it by M . By [Bro69, 4] or [Mey68, 4], all s § § degenerate triple systems are forms of one of these, meaning that any degenerate triple system becomes isomorphic to M over a separable closure of the ground field. s Example 1.5. For J an Albert F-algebra, there is a nondegenerate triple system denoted by M(J) whose underlying F-vector space is V = (F J ). Explicit formulas for the b, t, and J F q for this triple system can be found in [Bro69, 3], [Mey68, 6], [Fer72, 1], and [Gar99, 3.2]. § § § For Jd the split Albert F-algebra, we set Md := M(Jd). It is called the split triple system because Inv(Md) is the split simply connected algebraic group of type E over F [Gar99, 7 3.5]. By [Bro69, 4] or [Mey68, 4] every nondegenerate triple system is a form of Md. § § A similarity of triple systems is a map f : (V,b,t) ∼ (V′,b′,t′) defined by an F-vector −→ space isomorphism f: V ∼ V′ such that b′(f(x),f(y)) = λb(x,y) and t′(f(x),f(y),f(z)) = −→ λf(t(x,y,z)) for all x, y, z V and some λ F called the multiplier of f. Similarities with ∗ ∈ ∈ multiplier one are called isometries. They are the isomorphisms in the obvious category of Freudenthal triple systems. For a triple system M, we write Inv(M) for the algebraic group whose F-points are the isometries of M. Remark 1.6. Althoughitisnotclearprecisely what thestructureoftheautomorphismgroup of a degenerate triple system is, a few simple observations can be made which make it appear to be not very interesting from the standpoint of simple algebraic groups. Since by definition any element of Inv(M ) must preserve the quartic form q, it must s also be a similarity of the quadratic form det with multiplier 1. This defines a map ± Inv(M ) µ which is surjective since ̟ Inv(M )(F) maps to 1, where s 2 s −→ ∈ − α j β j ̟ := − ′ . j β j α ′ (cid:18) (cid:19) (cid:18) (cid:19) So Inv(M ) is not connected. s Also, we can make some bounds on the dimension. Specifically, we define a map f : G W GL(W) Inv(M ) by m,F s × × −→ α j cα φ(j) f(c,u,φ) := , j β αu+φ (j ) 1(β +s(φ(j),u)) (cid:18) ′ (cid:19) (cid:18) † ′ c (cid:19) where φ = σ(φ) 1 for σ the involution on End (W) which is adjoint with respect to s. (So † − F s(φ(w),φ (w )) = s(w,w) for all w,w W.) Then f is an injection of varieties, but † ′ ′ ′ ∈ f(c,u,φ)f(d,v,ψ) = f(cd,du+φ (v),φψ), † so it is not a group homomorphism. It does, however, restrict to be a morphism of algebraic groups on G 0 GL(W), so Inv(M ) contains a split torus of rank 28. This map f m,F s ×{ }× GROUPS OF TYPE E OVER ARBITRARY FIELDS 5 7 is also not surjective since for any u = 0, the map 6 α j 1(α s(φ(j ),u)) βu+φ (j) ̟f(c,u,φ)̟−1 = c − ′ † j β φ(j ) βc ′ ′ (cid:18) (cid:19) (cid:18) (cid:19) is not in the image of f. For an upper bound, we observe that the identity component of Inv(M ) is contained in the isometry group of det (whose identity component is of type D , s 28 hence is 1540-dimensional). Thus 757 < dimInv(M )+ 1540. s ≤ 2. An identity For a Freudenthal triple system (V,b,t) over F, we define an F-vector space map p : V V End (V) given by F F ⊗ −→ (2.1) p(u v)w := t(u,v,w) b(w,u)v b(w,v)u. ⊗ − − In the case where the triple system is nondegenerate, Freudenthal [Fre54, 4.2] also defined a map V V End (V) which he denoted by . The obvious computation shows that F F ⊗ −→ × his map is related to our map p by (2.2) 8v v = p(v v ). ′ ′ × ⊗ Theorem 2.3. Let M := (V,b,t) be a Freudenthal triple system with map p as given above. Then M is nondegenerate if and only if it satisfies the identity (2.4) tr(p(x x)p(y y)) = 24 q(x,x,y,y) 2b(y,x)2 ⊗ ⊗ − for all x,y V, where tr is the usual trace form on End (V). (cid:0) F (cid:1) ∈ Proof: If M is nondegenerate, then the conclusion is [Fre63, 31.3.1]orit can beeasily derived from [Mey68, 7.1]. So we may assume that M is degenerate and show that it doesn’t satisfy (2.4). Extending scalars, we may further assume that our ground field is separably closed and so that M = M , the degenerate triple system from Example 1.2. s To simplify some of our formulas, we define the weighted determinant, wdet: M F, s −→ to be given by wdet(x) := 3αβ s(j,j ) ′ − for x and y as in (1.4). We first compute the value of the left side of (2.4). For x and y as in (1.4), we can directly calculate the action of p(x x)p(y y) on each of the four entries of our matrix as in (1.3). ⊗ ⊗ Since we are interested in the trace of this operator, we only record the projection onto the entry that we are looking at. 1 0 (2.5) 4[wdet(x)wdet(y) 4αδs(j,k )] ′ 0 0 7→ − (cid:18) (cid:19) 0 0 (2.6) 4[wdet(x)wdet(y)+4βγs(j ,k)] ′ 0 1 7→ (cid:18) (cid:19) 6 R. SKIP GARIBALDI 0 m det(x)det(y)m+4(αδ s(k,j ))s(k ,m)j (2.7) 4 − ′ ′ 0 0 7→ +2det(x)s(k′,m)k +2det(y)s(j′,m)j (cid:18) (cid:19) (cid:20) (cid:21) 0 0 det(x)det(y)m 4(βγ s(j,k ))s(k,m)j (2.8) 4 ′ − − ′ ′ ′ m′ 0 7→ 2det(x)s(k,m′)k′ 2det(y)s(j,m′)j′ (cid:18) (cid:19) (cid:20) − − (cid:21) Since s is nondegenerate, it induces an identification of V with its dual V by sending ∗ x V to the map v s(x,v). We may also identify V V with End (V), and combining F ∗ F ∈ 7→ ⊗ these two identifications provides an isomorphism ϕs: V F V ∼ EndF(V) given by ⊗ −→ (2.9) ϕ (x y)w = xs(y,w), s ⊗ cf. [KMRT98, 5.1]. One has tr(ϕ (x y)) = s(y,x). s ⊗ With that notation, the terms in the brackets of (2.7) and (2.8) with coefficient 4 give the maps (αδ s(k,j ))ϕ (j k ) and (βγ s(j,k ))ϕ (j k) ′ s ′ ′ s ′ − ⊗ − − ⊗ on W, which have traces (αδ s(k,j ))s(k ,j) and (βγ s(j,k ))s(j ,k) ′ ′ ′ ′ − − respectively. Similarly, the terms with coefficient 2 give the maps det(x)ϕ (k k )+det(y)ϕ (j j ) and det(x)ϕ (k k) det(y)ϕ (j j) s ′ s ′ s ′ s ′ ⊗ ⊗ − ⊗ − ⊗ whose sum has trace 2[det(x)s(k ,k)+det(y)s(j ,j)]. ′ ′ Adding all of this up, we see that 1tr(p(x x)p(y y)) = wdet(x)wdet(y)+2βγs(j ,k) 2αδs(j,k ) 8 ⊗ ⊗ ′ − ′ +27det(x)det(y) (2.10) +2[(βγ s(j,k ))s(j ,k)+(αδ s(k,j ))s(k ,j)] ′ ′ ′ ′ − − +2[det(x)s(k ,k)+det(y)s(j ,j)] ′ ′ = 3q(x,x,y,y) b(y,x)2 +20det(x)det(y) 5det(x,y)2, − − where we have linearized the determinant to define det(x,y) := det(x+y) det(x) det(y). − − Taking the difference of the last expression in (2.10) and one-eighth of the right side of (2.4), we get the quartic polynomial (2.11) 5b(x,y)2 +20det(x)det(y) 5det(x,y)2. − We plug 0 j 0 k x := and y := j 0 k 0 ′ ′ (cid:18) (cid:19) (cid:18) (cid:19) into (2.11), where we have chosen j, j , k, and k such that ′ ′ s(j,k ) = s(j ,k) = 0 and s(j,j ) = s(k,k ) = 1. ′ ′ ′ ′ GROUPS OF TYPE E OVER ARBITRARY FIELDS 7 7 Then b(x,y) = 0 and det(x) = det(y) = 1. Since det(x+y) = 2, we have det(x,y) = 0. − − Thus the value of (2.11) is 20 and (2.4) does not hold for degenerate triple systems. It really was important that we allow x = y in (2.4), since for x = y (2.11) is identically 6 zero. So all triple systems satisfy the identity (2.12) tr(p(x x)2) = 24q(x,x,x,x). ⊗ 3. Gifts In this section we will define an object we call a gift, such that every adjoint group of type E is the automorphism group of some gift. We must first have some preliminary definitions. 7 Suppose that (A,σ) is a central simple algebra with a symplectic involution σ. The sandwich map Sand: A A End (A) F F ⊗ −→ defined by Sand(a b)(x) = axb for a,b,x A ⊗ ∈ is an isomorphism of F-vector spaces by [KMRT98, 3.4]. Following [KMRT98, 8.B], we § define a map σ on A A which is defined implicitly by the equation 2 F ⊗ Sand(σ (u))(x) = Sand(u)(σ(x)) for u A A, x A. 2 ∈ ⊗ ∈ Suppose now that A is split. Then A = End (V) for some F-vector space V, and σ is ∼ F the adjoint involution for some nondegenerate skew-symmetric bilinear form b on V (i.e., b(fx,y) = b(x,σ(f)y) for all f End (V)). As in (2.9), we have an identification ϕ : F b ∈ V F V ∼ EndF(V), and by a straightforward computation (or see [KMRT98, 8.6]), σ2 is ⊗ −→ given by (3.1) σ (ϕ (x x ) ϕ (x x )) = ϕ (x x ) ϕ (x x ) 2 b 1 2 b 3 4 b 1 3 b 2 4 ⊗ ⊗ ⊗ − ⊗ ⊗ ⊗ for x , x , x , x V. 1 2 3 4 ∈ Finally, for f: A A an F-linear map, we define f: A A A by F −→ ⊗ −→ f(a b) = f(a)b. ⊗ b Definition 3.2. A gift G over a fieldbF is a triple (A,σ,π) such that A is a central simple F-algebra of degree 56, σ is a symplectic involution on A, and π: A A is an F-vector −→ space map such that G1: σπ(a) = πσ(a) = π(a), − G2: aπ(a) = 2a2 for some a Skew(A,σ), 6 ∈ G3: π(π(a)a) = 0 for all a Skew(A,σ), ∈ G4: π σ Id = (π σ Id)σ , and 2 − − − − − G5: Trd (π(a)π(a)) = 24Trd (π(a)a) for all a,a (A,σ). A ′ A ′ ′ − ∈ b b b b b b 8 R. SKIP GARIBALDI By Skew(A,σ) we mean the vector space of σ-skew-symmetric elements of A, i.e., those a A such that σ(a) = a. ∈ − A definition as strange as 3.2 demands an example. Example 3.3. Suppose that M = (V,b,t) is a nondegenerate Freudenthal triple system over F. Set End(M) := (End (V),σ,π) where σ is the involution on End (V) adjoint to F F b. Using the identification ϕ : V V End (V), we define π: A A by π := pϕ 1, b ⊗ −→ F −→ −b where p is as in (2.1). We show that End(M) is a gift. A quick computation shows that b(π(ϕ (x y))z,w) = b(z,π(ϕ (x y))w) = b(π(ϕ (y x))z,w), b b b − ⊗ ⊗ − ⊗ which demonstrates G1, since σϕ (x y) = ϕ (y x). b b ⊗ − ⊗ Suppose that G2 fails. Then for v V, we set a := ϕ (v v) and observe that a2 = 0, so b ∈ ⊗ 0 = ϕ (v v)π(ϕ (v v))v = q(v,v,v,v)v. b b ⊗ ⊗ Since this holds for all v V, q is identically zero, contradicting FTS2. Thus G2 holds. ∈ Since elements a like in the preceding paragraph span Skew(A,σ), in order to prove G3 we may show that (3.4) π(π(a)a +π(a)a)y = 0, ′ ′ where (3.5) a = ϕ (x x) and a = ϕ (z z). b ′ b ⊗ ⊗ A direct expansion of the left-hand side of (3.4) shows that it is equivalent to FTS3. ′ Using just the bilinearity and skew-symmetry of b and the trilinearity oft, G4 is equivalent to t(x,y,x) = t(x,x,y) for all x,x,y V. ′ ′ ′ ∈ Thus G4 holds by FTS1. Finally, consider G5. If a is symmetric, then by G1 π(a) = 0 and the identity holds. If a ′ is symmetric then the left-hand side of G5 is again zero by G1. Since σ and Trd commute, A we have Trd (π(a)a) = σ(Trd (π(a)a)) = Trd (aπ(a)) = Trd (π(a)a), A ′ A ′ A ′ A ′ − − so the right-hand side of G5 is also zero. Consequently, by the bilinearity of G5, we may assume that a and a are skew-symmetric, and we may further assume that a and a are as ′ ′ given in (3.5). Then G5 reduces to (2.4). It turns out that this construction produces all gifts with split central simple algebra component. Lemma 3.6. Suppose that G = (A,σ,π) is a gift over F. Then G = End(M) for some ∼ nondegenerate Freudenthal triple system over F if and only if A is split. GROUPS OF TYPE E OVER ARBITRARY FIELDS 9 7 Proof: One direction is done by Example 3.3, so suppose that (A,σ,π) is a gift with A split. Then we may write A = End (V) for some 56-dimensional F-vector space V such that V is ∼ F endowed with a nondegenerate skew-symmetric bilinear form b and σ is the involution on A which is adjoint to b. We define t: V V V V by × × −→ t(x,y,w) := π(ϕ (x y))w+b(w,x)y +b(w,y)x. b ⊗ Observe that t is trilinear. We define a 4-linear form q on V as in FTS2. The proof that FTS3 implies G3 in Example 3.3 reverses to show that G3 implies ′ FTS3. Similarly, G4 implies that t(x,y,z) = t(x,z,y) for all x,y,x V, so q(w,x,y,z) = ′ ′ ∈ q(w,x,z,y). G1 implies that q(w,x,y,z) = q(z,x,y,w) = q(w,y,x,z). Since the trans- positions (34), (14), and (23) generate (= the symmetric group on four letters) q is 4 S symmetric. Next, suppose that FTS2 fails, so that q is identically zero. Then since b is nondegenerate, t is also zero. Then for v,v ,z V and a := ϕ (v v) and a := ϕ (v v ), ′ b ′ b ′ ′ ∈ ⊗ ⊗ (aπ(a)+aπ(a))z = 2(b(v,v )b(v ,z)v +b(v ,v)b(v,z)) = 2(aa +aa)z. ′ ′ ′ ′ ′ ′ ′ Since elements of the same form as a and a span Skew(A,σ), this implies that G2 fails, ′ which is a contradiction. Thus FTS2 holds and (V,b,t) is a Freudenthal triple system. Finally, writing out G5 in terms of V gives (2.4), which shows that (V,b,t) is nondegen- erate. Remark 3.7. The astutereader willhave noticed thatour definitionof End(M) almost works if M is degenerate, in that the only problem is that the resulting (A,σ,π) doesn’t satisfy G5. That example and the proof of 3.6 make it clear that if we remove the axiom G5 from the definition of a gift, then we would get an analog to Lemma 3.6 where the Freudenthal triple system is possibly degenerate. Remark 3.8. Observe that in the isomorphism G = End(M) from the preceding lemma, M ∼ is only determined up to similarity. To wit, for a nondegenerate triple system M = (V,b,t) and λ F , we define a similar structure M = (V,λb,λt). Then M is also a nondegenerate ∗ λ λ ∈ triple system and End(M) = End(M ). The only potential difficulty with this last equality λ would be if the π produced by M , which we shall denote by π , is different from the π λ λ produced by M. However, we see that π (ϕ (x y))w = λt(x,y,w) λb(w,x)y λb(w,y)x λ λb ⊗ − − = λπ(ϕ (x y))w b ⊗ = π(ϕ (x y))w. λb ⊗ Isometries and derivations. Definition 3.9. An isometry in a gift G := (A,σ,π) is an element f A such that σ(f)f = ∈ 1 and π(faf 1) = fπ(a)f 1 for all a A (this ensures that Int(f) is an automorphism − − ∈ of G). The motivation for the name is that then the isometries in End(M) are just the isometries of M. We set Iso(G) to be the algebraic group whose F-points are the group of isometries in G. 10 R. SKIP GARIBALDI A derivation in G is an element f Skew(A,σ) which satisfy ∈ GD: π(fa) π(af) = fπ(a) π(a)f for all a A. − − ∈ We define Der(G) to be the vector space of derivations in G. The name derivation has the same sort of motivation: We will see in the proof of Proposition 3.11 that the derivations in End(V,b,t) are precisely the maps in End (V) which are traditionally called derivations of F the triple system (V,b,t). 3.10. The description of the isometries in End(V,b,t) combined with Lemma 3.6 shows that Iso(G) is simple simply connected of type E . Since any automorphism of G is also an 7 isomorphism of A, it must be of the form Int(f) for some f A such that σ(f)f F . ∗ ∗ ∈ ∈ Thus the map Iso(G) Aut(G) is a surjection over a separable closure of the ground −→ field. Since the kernel of this map is the center of Iso(G), Aut(G) is simple adjoint of type E . 7 We can actually say more. The group Iso(G) is a subgroup of the symplectic group Sp(A,σ), whose F-points are the elements f A such that σ(f)f = 1. It is this embedding ∈ combined with the vector representation of Sp(A,σ) (= the natural embedding of Sp(A,σ) in A ) which gives rise to the Tits algebra of Iso(G), and so the Tits algebra of Iso(G) is ∗ the same as the Tits algebra of Sp(A,σ), which is just A. It is easy to see that Der(G) is actually a Lie subalgebra of Skew(A,σ), where the bracket is the usual commutator. In fact, by formal differentiation as in [Bor91, 3.21] or [Jac59, 4], § Der(G) is the Lie algebra of Iso(G). Proposition 3.11. For G := (A,σ,π) a gift, imπ = Der(G). Proof: Since imπ and Der(G) are both vector subspaces of A, it is equivalent to prove this over a separable closure. Thus we may assume that A is split, so that G = End(M) for some nondegenerate triple system M := (V,b,t) over F by Lemma 3.6. Consider the vector subspace D of Skew(A,σ) consisting of elements d such that dt(u,v,w) = t(du,v,w)+t(u,dv,w)+t(u,v,dw) for all u,v,w V. (These are known as the derivations of M.) The obvious computation ∈ shows that imπ D, which one can find in [Mey68, p. 166, Lem. 1.3]. Conversely, the ⊆ reverse containment holds by [Mey68, p. 185, S. 8.3]. (He has an “extra” hypothesis that the characteristic of F is not 5 because he is also considering triple systems of dimensions 14 and 32, but that is irrelevant for our purposes.) For d Skew(A,σ), consider the element ∈ (3.12) π(da) π(ad) dπ(a)+π(a)d − − in A, which is zero if and only if d satisfies GD. Since π is linear, 3.12 is zero for all a if and only if it is zero when a = ϕ (u v). Applying the endomorphism 3.12 to w V and b ⊗ ∈ expanding out, we obtain t(du,v,w)+t(dv,u,w)+t(u,v,dw) dt(u,v,w). − So d satisfies GD if and only if it lies in D = imπ.