Sextonions, Zorn Matrices, and E 1 72

DFPD/2015/TH/10

Sextonions, Zorn Matrices, and e 1 72

Alessio Marrani 1,2 and Piero Truini 3

1 Centro Studi e Ricerche “Enrico Fermi”, Via Panisperna 89A, I-00184, Roma, Italy

2 Dipartimento di Fisica e Astronomia “Galileo Galilei”, Universit`adi Padova, and INFN, sezione di Padova, Via Marzolo 8, I-35131 Padova, Italy [email protected]

3 Dipartimento di Fisica, Universit`adegli Studi and INFN, sezione di Genova, via Dodecaneso 33, I-16146 Genova, Italy [email protected]

ABSTRACT

By exploiting suitably constrained Zorn matrices, we present a new construction of the algebra of sextonions (over the algebraically closed ﬁeld C). This allows for an explicit construction, arXiv:1506.04604v3 [math.RA] 20 May 2017 in terms of Jordan pairs, of the non-semisimple Lie algebra e 1 , intermediate between e7 and 7 2 e8, as well as of all Lie algebras occurring in the sextonionic row and column of the extended Freudenthal Magic Square. MSC(2010) numbers: 17B10, 17B25, 17B45. Keywords : exceptional Lie algebras, intermediate algebras, sextonions, Zorn matrices. 1 Introduction

The field of composition, non-associative algebras, and related Lie algebras, underwent a series of interesting developments in recent times. In [1] Deligne proposed dimension formulas for the exceptional series of complex simple Lie algebras, whose parametrization in terms of the dual Coxeter number was exploited further in [2] by Cohen and de Man (see also [3]). Landsberg and Manivel subsequently pointed out the relation between the dimension formulas and the dimensions of the composition algebras themselves in [4]. In [1, 2] it was observed that all parameter values determining integer outputs in the dimension formulas were already accounted for by the known normed division algebras, with essentially one exception, intriguingly corresponding to a would-be composition algebra of dimension six, sitting between the quaternions and octonions. This algebra, whose elements were named sextonions, was recently studied by Westbury in [5], who pointed out the related existence of a whole new row in the Freudenthal Magic Square. Actually, the six-dimensional algebra of sextonions had been observed earlier as a curiosity; indeed, it was explicitly constructed in [6]. Moreover, it was used in [7] to study the conjugacy classes in the smallest exceptional Lie algebra g2 in characteristics other than 2 or 3. The sextonions were also constructed in [8] (cfr. Th. 5 therein), and proved to be a maximal subalgebra of the split octonions. In [9], Landsberg and Manivel “filled in the hole” in the exceptional series of Lie algebras, observed by Cvitanovic, Deligne, Cohen and de Man, showing that sextonions, through the triality construction of [4], give rise to a non-simple intermediate exceptional Lie algebra, named e 1 , between e7 and e8, satisfying some of the decomposition and dimension formulas 7 2 of the exceptional simple Lie algebras [1, 2, 3, 4, 10]. More recently, such a 190-dimensional Lie algebra e 1 was also found by Mkrtchyan in the 7 2 study of the Vogel plane [11], in the context of the analysis of the universal Vogel Lie algebra [12]. By the Hurwitz Theorem [13], the real normed division algebras are the real numbers R, the complex numbers C, the quaternions H and the octonions C (Cayley numbers). Each algebra can be constructed from the previous one by the so-called Cayley-Dickson doubling procedure [14, 15]. All these algebras can be complexified to give complex algebras. These complex algebras respectively are R ⊗ C = C, C ⊗ C = C ⊕ C, H ⊗ C = M2(C), C ⊗ C (M2 denoting a 2 × 2 matrix). The three complex algebras other than C have a second real form, denoted Cs, Hs and Cs, with the following isomorphisms holding : Cs = R ⊕ R and Hs = M2(R). The normed division algebras are called the compact forms and the aforementioned second real form is called the split real form. It is worth pointing out that split real forms are composition algebras but they are not division algebras. On the field R, the sextonions only exist in split form Ss, and they are intermediate between the split quaternions Hs and the split octonions Cs :

Hs ⊂ Ss ⊂ Cs. (1.1)

Note that Ss does not contain the divisional quaternions H; see App. A. Nowadays, exceptional Lie algebras have a long-standing history of applications to physics (see e.g. [16], [17], [18]-[25] for a partial list of results and Refs.). The relevance of compact exceptional Lie algebras (and groups) in realizing grand uniﬁcation gauge theories and consistent string theories is well recognized. Similarly, the relevance of non-compact real forms for

1 the construction of locally supersymmetric theories of gravity is well appreciated. Other frame- works include sigma models based on quotients of exceptional Lie groups, which are of interest for string theory and conformal ﬁeld theories, as well. It is here worth pointing out that that the analysis of quantum criticality in Ising chains and the structure of magnetic materials such as Cobalt Niobate has also recently (and strikingly) turned out to be related to exceptional Lie algebras of type E (see e.g. [26] and [27], respectively). Moreover, exceptional Lie algebras occur in models of conﬁnement in non-Abelian gauge theories (for instance, cfr. [21]), as well as in a striking relation between cryptography and black hole physics, recently discovered [29]-[34]. It should also be recalled that fascinating exceptional algebraic structures arise in the description of the Attractor Mechanism for black holes in Maxwell-Einstein supergravity theories [35]-[41], such as the so-called magic exceptional supergravity [42]-[46]. In this context, the aforementioned, intermediate 190-dimensional Lie algebra e 1 is quite 7 2 novel, and applications to physics are still under investigation, even though recent studies (cfr. e.g. [47]) intriguingly seem to connect sextonions to theories beyond eleven-dimensional M- theory. It should also be mentioned that e 1 can be regarded as a Freudenthal triple system 7 2 O over the exceptional Albert algebra J3 , along with its automorphism algebra e7 and an extra e7-singlet generator, acting on the Freudenthal triple system as the multiplication by a scalar; for details on the applications of Freudenthal triple systems to the study of black hole attractors in four dimensions, cfr. e.g. [40, 48, 49, 50], and Refs. therein.

In the present paper we will apply the formal machinery introduced in [51] and [52], as well as an explicit realization of the sextonions (over the algebraically closed ﬁeld C), in order to explicitly construct the non-semisimple Lie algebra e 1 , as well as all algebras occurring in the 7 2 sextonionic row of the extended Freudenthal Magic Square [5, 9], in terms of Jordan pairs.

The plan of the paper is as follows. In Sec. 2 we provide a realization of the sextonions in terms of nilpotents constructed from the traceless octonions, and recall their representation in terms of suitably constrained Zorn matrices. The intermediate exceptional algebra e 1 is then considered in Sec. 3, which focuses on 7 2 the construction (then developed in Secs. 6 and 7) of the sextonionic row and column of the extended Magic Square, by exploiting Jordan pairs for the sextonionic rank-3 Jordan algebra. The action of g2 = Der(C) on the Zorn matrices is recalled in Sec. 4, and exploited in Sec. 5 to determine the derivations of the sextonions, Der(S). Then, in Secs. 6 and 7 the explicit construction of the intermediate algebras c 1 (which 3 2 analogously holds for a 1 and d 1 ) and e 1 is presented. 5 2 6 2 7 2 The paper is concluded by App. A, in which we prove that, on the ﬁeld R, Ss does not contain the divisional quaternions H.

2 Sextonions and their Nilpotent Realization

The algebra of sextonions is a six dimensional subalgebra S of the octonions. As mentioned above, we denote by C the algebra of the octonions over the complex ﬁeld C, whose multiplication rule goes according to the Fano diagram in Figure 1. P7 If a ∈ C we write a = a0 + j=1 ajuj where aj ∈ C for j = 1,..., 7 and uj for j = 1,..., 7 denote the octonion imaginary units. We denote by i the the imaginary unit in C. We introduce 2 idempotent elements: 1 ρ± = (1 ± iu ) 2 7

2 Figure 1: Fano diagram for the octonions’ products and 6 nilpotent elements : ± ± εk = ρ uk , k = 1, 2, 3 One can readily check that [51]:

(ρ±)2 = ρ± , ρ±ρ∓ = 0

± ± ± ∓ ± ρ εk = εk ρ = εk

∓ ± ± ± ρ εk = εk ρ = 0

± 2 (εk ) = 0 (2.1)

± ± ± ± ∓ εk εk+1 = −εk+1εk = εk+2 (indices modulo 3)

± ∓ εj εk = 0 j 6= k

± ∓ ± εk εk = −ρ

+ + − − + + − − We can write a ∈ C as a = α0 ρ + α0 ρ + αk εk + αk εk . ± ± + − − + The subalgebra S ∈ C generated by ρ , ε1 , ε2 , ε3 (namely a ∈ S iﬀ α2 = α3 = 0) provides an explicit realization of the sectonions. The existence of the non-divisional sextonionic elements can be easily understood. Indeed, in order to construct divisional sextonions, one would need to combine a nilpotent with its complex conjugate; but, as given by the above construction, + − this is not possible for ε2 nor for ε3 . Octonions can be represented by Zorn matrices [53]. After the treatment of Sec. 3 of [52], we can represent the sextonions as a Zorn matrix, as long as A+ and A− are C3-vectors of the type A+ = (a+, c+, 0) and A− = (a−, 0, c−) Notice that A+ and A− lie on orthogonal C3-planes sharing the line along the ﬁrst component.

3 e 1 72

In recent papers [51, 52] a unifying view of all exceptional Lie algebras in terms of a2 subalgebras and Jordan Pairs has been presented, and a Zorn-matrix-like representation of these algebras has been introduced. The root diagram related to this view is shown in Figure 2, where the roots of the exceptional Lie algebras are projected on a complex su(3) = a2 plane, recognizable by the dots forming the

3 Figure 2: A unifying view of the roots of exceptional Lie algebras through Jordan pairs [51]. For n = 8, the root diagram of e8 is obtained.

Figure 3: Root diagram of e 1 (for n = 8) 7 2 external hexagon, and it exhibits the Jordan pair content of each exceptional Lie algebra. There n n are three Jordan pairs (J3 , J3 ), each of which lies on an axis symmetrically with respect to the n center of the diagram. Each pair doubles a simple Jordan algebra of rank 3, J3 , with involution - n the conjugate representation J3 , which is the algebra of 3×3 Hermitian matrices over A, where A = R, C, H, C for n =dimRA = 1, 2, 4, 8 respectively, stands for real, complex, quaternion, octonion algebras, the four normed division algebras according to Hurwitz’s Theorem; see e.g. [54]. Exceptional Lie algebras f4, e6, e7, e8 are obtained for n = 1, 2, 4, 8, respectively. g2 (corresponding to n = −2/3) can be also represented in the same way, with the Jordan algebra reduced to a single element. For further detail, cfr. [51]. We expand that view in this paper to include e 1 [9], a Lie subalgebra of e8 of dimension 7 2 190. If we consider the e8 root diagram (obtained in Figure 2 for n = 8), then the sub-diagram of e 1 is shown in Figure 3, (for n = 8, as well). 7 2 In general, one can do the same for all algebras in the fourth and third row of the Magic square, [55] [56], that we denote by gIV and gIII respectively (see table 1). In this way, the algebras in the intermediate (fourth) row of the extended Magic Square [5, 9] are explicitly

4 Figure 4: Diagram of g 1 III 2 constructed in terms of Jordan pairs.

n 1 2 4 8

gIII c3 a5 d6 e7

gIV f4 e6 e7 e8

Table 1: Third and fourth row of the magic square

We get a subalgebra of gIV , that we denote here by g 1 , given by gIII plus a (6n + 8)- III 2 1 dimensional irreducible representation of gIII plus a gIII-singlet, as shown in Figure 4 . In particular, the irreps. of gIII are symplectic (i.e., they admit a skew-symmetric invariant form), and they have complex dimension 6n + 8 = 14, 20, 32, 56 for n = 1, 2, 4, 8 respectively; the algebras g 1 are their corresponding Heisenberg algebras (denoted by H) through such an III 2 invariant tensor, [9], c 1 = c3•H14, a 1 = a5•H20, d 1 = d6•H32, e 1 = e7•H56, of complex 3 2 5 2 6 2 7 2 dimension 36, 56, 99, 190.

Let us here present a brief account of the Jordan pairs for sextonions S by means of suitable embeddings. We start with the maximal, non-symmetric embedding:

e7 ⊃ a2 ⊕ a5 (3.1) 133 = (8, 1) + (1, 35) + (3, 15) + 3, 15 (3.2) 56 = (3, 6) + 3, 6 + (1, 20) , (3.3) implying that: e7 n 56 ⊃ [a2 ⊕ (a5 n 20)] n (3, 15 + 6) + 3, 15 + 6 . (3.4) n=6 Thus, the Jordan pairs for the sextonionic Jordan algebra of rank 3, J3 , are given by (3, 15 + 6) + 3, 15 + 6 in (3.4). In order to reconstruct the extended Magic Square [5, 9], one needs also to add the extra column shown in table 2, [9], where a further algebra d 1 1 = d6•H32•H44 is introduced. 6 2 2 1There are some variations on the deﬁnition of intermediate algebra, [5] [9], based on the grading induced by 00 an highest root. Our realisation of Der( ) and e 1 corresponds to the algebra denoted by g in the Introduction S 7 2 of [9].

5 n 6

gI c 1 3 2 gII a 1 5 2 gIII d 1 6 2

g 1 d6 1 1 III 2 2 2 gIV e 1 7 2

Table 2: Sixth column of the magic square

6 This column corresponds to the Jordan algebra that we denote by J3 of 3 × 3 Hermitian matrices over the sextonions. The new element d6•H32•H44 can be easily seen in the diagram 6 6 of ﬁgure 4 for n = 6: g = a 1 is the reduced structure algebra of J , gIII = d 1 the 0 5 2 3 6 2 6 super-structure algebra of J and ﬁnally d 1 1 = d 1 •H44 = d6•H32•H44. Notice that the 3 6 2 2 6 2 6 6 44-dimensional representation of d 1 is made of J ⊕ J ⊕ 2. Finally, the algebra e 1 at the 6 2 3 3 7 2 6 end of the column is viewed as in the diagram of Figure 2 for n = 6, with g = a 1 and the 0 5 2 subalgebra e7 represented by the same diagram for n = 4. This completes the explicit construction of the relevant rows and columns (pertaining to the sextonions) of the extended Magic Square2 [5, 9].

4 g2 action on Zorn matrices In our previous paper [52], we have introduced the following adjoint representation % of the Lie algebra g2: a A+ (4.1) A− 0 + − 3 where a ∈ a2, A ,A ∈ C , viewed as column and row vector respectively. The commutator of two such matrices reads [52]:

a A+ b B+ , A− 0 B− 0 (4.2) [a, b] + A+ ◦ B− − B+ ◦ A− aB+ − bA+ + 2A− ∧ B− = A−b − B−a + 2A+ ∧ B+ 0 where A+ ◦ B− = t(A+B−)I − t(I)A+B− (4.3) (with standard matrix products of row and column vectors and with I denoting the 3 × 3 identity matrix); A ∧ B is the standard vector product of A and B, and t(a) denotes the trace of a.

The g2 generators are [51]:

± %(d√k ) = Ek±1 k±2 (mod 3) ,√ k = 1, 2, 3 %( 2H1) = E11 − E22 %( 6H2) = E11 + E22 − 2E33 (4.4) + + − − %(gk ) = Ek4 := ek %(gk ) = E4k := ek , k = 1, 2, 3

2It is once again worth stressing that in the present investigation, as well as in the previous papers [51, 52], we only consider complex forms of the Lie algebras.

6 Figure 5: Diagram of g2 with corresponding generators and matrix-like elements

where Eij denotes the matrix with all zero elements except a 1 in the {ij} position: (Eij)k` = + 3 − δikδj` and ek are the standard basis vectors of C (ek are their transpose). The correspondence with the roots of g2 is shown in Figure 5.

5 Derivations of S

We now use the representation % to get a representation of the Lie algebra of Der(S), which indeed is a non-reductive subalgebra of g2 = Der(C). It was shown in [5] that the map from the subalgebra of derivations of C preserving S, that we here denote by DerC(S), to Der(S) is surjective with one-dimensional kernel; the corresponding statement at the level of automorphism group was made in [9]. Within our formalism, this result is achieved by restricting %(g2) to the matrices that preserve S. One easily gets:  a 0 a   s+  a S+ 11 13 1 : a = a a a ,S+ = s+ ,S− = (s−, 0, s−) (5.1) S− 0  21 22 23   2  1 3 0 0 a33 0

+ We also realize very easily that the generator corresponding to d1 , namely the element E23 in %(g2), acts trivially on S, hence it can be set to 0. The commutator (4.2) must be modiﬁed accordingly, by setting the {23} element of a equivalent to zero, that is by replacing the standard matrix product of two matrices     a11 0 a13 b11 0 b13 a =  a21 a22 0  , b =  b21 b22 0  (5.2) 0 0 a33 0 0 b33 with the new product a b = ab − E22 ab E33 (5.3) and the product S+ ◦ S− with

+ − + − + − + − S ◦ S = t(S S )I − t(I)(S S − E22S S E33) (5.4)

7 Figure 6: Root diagram of Der(S)

3 We thus have Der(S) = a1 ⊕ C ⊕ V4, where V4 is a 4-dimensional (spin-3/2) irreducible representation of a1 (as conﬁrmed by the entry in the ﬁrst column, fourth row in the extended Magic Square; cfr. e.g. [9]). The corresponding root diagram is shown in Figure 6, where we have also included the axes corresponding the linear span of the Cartan generators, represented by the matrices:

 −2 0 0   0 0 0  h 0 1,2 : h = 0 1 0 , h = 0 1 0 (5.5) 0 0 1   2   0 0 1 0 0 −1 Proposition 5.1 : The algebra spanned by the generators corresponding to the roots in Fig.6 is a Lie algebra. − − + − Proof : By looking at the diagram in Fig.5 these generators are d2 , d3 , g2 , g3 spanning a ± subspace L1 of g2, plus the generators g1 , h1, h2 spanning the Lie subalgebra L0 := a1 ⊕ C. We + have [L0,L0] ⊂ L0,[L0,L1] ⊂ L1,[L1,L1] ⊂ L2 ∼ 0, where L2 is the span of d1 . The notation is that of the grading with respect to h2. + We consider the g2 commutation relations among these generators and identify d1 ∼ 0. We only need to prove that the Jacobi identity is consistent with this identiﬁcation. Let X,Y,Z ∈ L0 ⊕ L1, then consistency must be checked in only two cases (up to cyclic permutation): + case 1: [X,Y ] ∝ d1 ; + case 2: [[X,Y ],Z] ∝ d1 . + Case 1: Consistency requires [[Y,Z],X] + [[Z,X],Y ] ∼ 0. This is true if [d1 ,Z] = 0, since + it is true in g2. On the other hand, if [d1 ,Z] 6= 0 then Z ∝ h2 and [Z,X] = λX ,[Z,Y ] = λY + ,since X,Y must be in L1 by hypothesis. Therefore [[Y,Z],X]+[[Z,X],Y ] = 2λ[X,Y ] ∝ d1 ∼ 0. Case 2: Both [X,Y ] and Z must be in L1. In particular either X or Y must be in L1. Sup- pose X ∈ L1. Then [Y,Z] ∈ L1 hence we have both [X,Z] ∼ 0 and [[Y,Z],X] ∼ 0. Similarly if Y ∈ L1.

This concludes the proof 3 This representation also characterizes a1 as the smallest Lie group “of type E7” [57], and it pertains to the so-called T 3 model of N = 2, D = 4 supergravity.

8 6 n = 1 : Matrix representation of c 1 32

1 We denote by a dot the Jordan product x·y = 2 (xy + yx) and by t() the ordinary trace of 3 × 3 1 2 matrices. We also set t(x, y) := t(x·y). For J3 and J3, obviously t(x, y) = t(xy). We use in this section the representation % of f4 in the form of a matrix introduced in [52], restricted to the subalgebra c 1 : 3 2 + a ⊗ I + I ⊗ a1 s %(f) = − T (6.1) s −I ⊗ a1 where    +  a11 0 a13 s1 + + − − − a =  a21 a22 0  , t(a) = 0 , s =  s2  , s = (s1 , 0, s3 ) (6.2) 0 0 a33 0 T ± 1 and a1 ∈ a2, a1 is the transpose of a1, I is the 3 × 3 identity matrix, si ∈ J3 , i = 1, 2, 3. The commutator is set to be:

+ + a ⊗ I + I ⊗ a1 s b ⊗ I + I ⊗ b1 r − T , − T s −I ⊗ a1 r −I ⊗ b1 (6.3) C C := 11 12 C21 C22 where, denoting by [a b] the commutator with respect to the product (5.3)

[a b] = a b − b a = [a, b] − E22[a, b]E33 (6.4) , it holds that:

+ − + − C11 = [a b] ⊗ I + I ⊗ [a1, b1] + s r − r s

+ + + + T C12 = (a ⊗ I)r − (b ⊗ I)s + (I ⊗ a1)r + r (I ⊗ a1 ) + + T − − − (I ⊗ b1)s − s (I ⊗ b1 ) + s × r (6.5) − − T − − C21 = −r (a ⊗ I) + s (b ⊗ I) − (I ⊗ a1 )r − r (I ⊗ a1) T − − + + + (I ⊗ b1 )s + s (I ⊗ b1) + s × r

T T − + − + C22 = I ⊗ [a1 , b1 ] + s • r − r • s with the following deﬁnitions (summing over repeated indices) :

+ − 1 + − + − s r := 3 t(s1 , r1 )I − (1 − (E23)ij)t(si , rj )Eij ⊗ I+ 1 + − + − I ⊗ 3 t(s1 , r1 )I − s1 r1

s− • r+ := I ⊗ ( 1 t(s−, r+)I − s−r+) 3 1 1 1 1 (6.6)

± ± ± ± ± ± ± ± ± ± (s × r )i := ijk[sj rk + rk sj − sj t(rk ) − rk t(sj ) ± ± ± ± − (t(sj , rk ) − t(sj )t(rk ))I] ± ± := ijk(sj #rk ) Notice that:

1 1. s ∈ J3 is a symmetric complex matrix;

9 + − 2. writing s r := c ⊗ I + I ⊗ c1 we have that both c and c1 are traceless hence c ia a − + T matrix like a in (5.1), c1 ∈ a2 and r • s = I ⊗ c1

+ + T 3 1 3. terms like (I ⊗ a1)r + r (I ⊗ a1 ) are in C ⊗ J3, namely they are matrix valued vectors with symmetric matrix elements;

1 ± ± 4. the sharp product # of J3 matrices appearing in s × r is a fundamental product in the theory of Jordan Algebras, [54]. It is the linearization of x# := x2 − t(x)x − 1 2 2 2 (t(x ) − t(x) )I, in terms of which we may write the fundamental cubic identity for n J3 , n = 1, 2, 4, 8: 1 1 x# ·x = t(x#, x)I or x3 − t(x)x2 + t(x#)x − t(x#, x)I = 0 (6.7) 3 3

3 2 8 2 2 where x = x ·x (notice that for J3, because of non-associativity, x x 6= xx in general). The validity of the Jacobi identity for the algebra of matrices (6.1) with Lie product given by (6.3) - (6.6) derives from the Jacobi identity for ρ(f4) proven in [52] together with Proposition 5.1, applied to c 1 by trivially extending the three grading argument. The validity of the Jacobi 3 2 identity, together with the fact that the representation % fulﬁlls the root diagram of c 1 (as 3 2 can be easily seen) proves that % is indeed a representation of c 1 . 3 2 Before passing to e 1 , let us point out that the cases of a 1 (n = 2) and d 1 (n = 4) can 7 2 5 2 6 2 be worked out in the same fashion as for c 1 , starting from the representations of e6 and e7 3 2 introduced in [52].

7 n = 8 : Matrix representation of e 1 72

We recall a few concepts and notations from [52]. We use the notation Lxz := x · z and, 3 8 3 for x ∈ ⊗ J with components (x , x , x ), L ∈ ⊗ L 8 denotes the corresponding C 3 1 2 3 x C J3 operator valued vector with components (Lx1 ,Lx2 ,Lx3 ). We can write an element a1 of e6 P 8 as a1 = Lx + [Lxi ,Lyi ] where x, xi, yi ∈ J3 and t(x) = 0. The adjoint is deﬁned by † a1 := Lx − [Lx1 ,Lx2 ]. Notice that the operators F := [Lxi ,Lyi ] span the f4 subalgebra of e6, 8 8 the derivation algebra of J3 . (Recall that the Lie algebra of the structure group of J3 is e6 ⊕C.)

† 8 8 We remark that (a1, −a1) is a derivation in the Jordan Pair (J3, J3), and it is useful to recall that the relationship between the structure group of a Jordan algebra J and the automorphism −1 group of a Jordan Pair V = (J, J) goes as follows, [58]: if g ∈ Str(J) then (g, Ug(I)g) ∈ Aut(V ). In our case, for g = 1 + (Lx + F ), at ﬁrst order in (namely, in the tangent space of the −1 2 corresponding group manifold) we get Ug(I)g = 1 + (−Lx + F ) + O( ).

Next. we introduce a product ? such that Lx ?Ly := Lx·y + [Lx,Ly], F?Lx := 2FLx and 3 8 3 8 Lx ?F := 2LxF for each component x of x ∈ C ⊗ J3 and y of y ∈ C ⊗ J3. If we denote by [; ] the commutator with respect to the ? product, we also require that [F1; F2] := 2[F1,F2]. We have that, Lx ?Ly + Ly ?Lx = 2Lx·y and [F ; Lx] := F?Lx − Lx ?F = 2[F,Lx] = 2LF (x), where 8 the last equality holds because F is a derivation in J3.

Therefore, for f ∈ e 1 , we write: 7 2 a ⊗ Id + I ⊗ a1 Ls+ %(f) = † (7.1) Ls− −I ⊗ a1

10 ± where a, s are the same as in (5.1), a1 ∈ e6, I is the 3 × 3 identity matrix, Id := LI is the identity operator in L 8 : L L = L . Notice that Id is the identity also with respect to the ? J3 I x x product. By extending the ? product in an obvious way to the matrix elements (7.1), one achieves † † + + + − − † that (I ⊗ a1) ?Lr + Lr ? (I ⊗ a1) = 2L(I⊗a1)r and (I ⊗ a1) ?Lr + Lr ? (I ⊗ a1) = 2L − . (I⊗a1)r After some algebra, the commutator of two matrices like (7.1) can be computed to read : a ⊗ Id + I ⊗ a1 Ls+ b ⊗ Id + I ⊗ b1 Lr+ † , † Ls− −I ⊗ a1 Lr− −I ⊗ b1 (7.2) C C := 11 12 , C21 C22 where: C11 = [a b] ⊗ Id + 2I ⊗ [a1, b1] + Ls+ Lr− − Lr+ Ls−

+ + + C12 = (a ⊗ Id)Lr − (b ⊗ Id)Ls + 2L(I⊗a1)r + − − − 2L(I⊗b1)s + Ls × Lr (7.3) C21 = −Lr− (a ⊗ Id) + Ls− (b ⊗ Id) − 2L † − (I⊗a1)r + 2L † − + Ls+ × Lr+ (I⊗b1)s

† † C22 = 2I ⊗ [a1, b1] + Ls− • Lr+ − Lr− • Ls+ . The products in (7.3) are deﬁned as follows :

1 + − + − Ls+ Lr− := 3 t(s1 , r1 )I − (1 − (E23)ij)t(si , rj )Eij ⊗ Id+ 1 + − I ⊗ t(s , r )Id − L + − − [L + ,L − ] 3 1 1 s1 ·r1 s1 r1

1 − + (7.4) Ls− • Lr+ := I ⊗ ( t(s , r )Id − L − + − [L − ,L + ]) 3 1 1 s1 ·r1 s1 r1

Ls± × Lr± := Ls±×r± = L ± ± ijk(sj #rk )

From the properties of the triple product of Jordan algebras, it holds that L + − +[L + ,L − ] = s1 ·r1 s1 r1 1 † † † V + − ∈ e6 ⊕ , [52]. Moreover one can readily check that [a , b ] = −[a1, b1] and Lr− • Ls+ = 2 s1 ,r1 C 1 1 1 + − † I ⊗ ( t(s1 , r1 )Id − Ls+·r− − [Ls+ ,Lr− ]) ; this result implies that we are actually considering an 3 1 1 1 1 algebra. The validity of the Jacobi identity for the algebra of matrices (7.1) with Lie product given 4 by (7.2) - (7.4) derives from the Jacobi identity for ρ(e8) (proven in [52]), together with Proposition 5.1, applied to e 1 by trivially extending the three grading argument. That the 7 2 Lie algebra so represented is e 1 is made obvious by a comparison with the root diagram in 7 2 ﬁgure 3.

A Real Forms

We use the notations of [52]. From the treatment in [52], a real form of octonions is obtained ± ± ± ± by taking α0 , αk ∈ R. The quaternionic subalgebra generated by ρ , ε1 is obviously a split 4We would like to recall that the proof of the Jacobi identity given in [52] strongly relies on identities deriving from the Jordan Pair axioms [58].

11 ± form with nilpotent ε1 .

Another real form is obtained by taking complex coeﬃcients with complex conjugation denoted by ‘∗’ subject to the conditions:

− + ∗ − + ∗ − + ∗ α0 = (α0 ) , α1 = −(α1 ) , α3 = (α2 ) (A.1)

Its quaternionic subalgebra, generated by 1, u7, iu1, iu4, is also split with nilpotent u7 +iuk , k = 1, 4. It is equivalent to the one obtained with all real coeﬃcients, which is generated by 1, u1, iu4, iu7 upon cyclic permutation of the indices 7, 4, 1.

Let us now restrict the Zorn matrix product [52] to the sextonions and introduce the vectors

E1 = (1, 0, 0) E2 = (0, 1, 0) E3 = (0, 0, 1) We get:

+ + + + α0 A β0 B − − − − A α0 B β0

+ + + − + + − + + α0 β0 − α1 β1 (α0 β1 + β0 α1 )E1 = − − + − − − − − + (A.2) (α0 β1 + β0 α1 )E1 α0 β0 − α1 · β1

+ + − + − − − − + 0 (α0 β2 + β0 α2 + α3 β1 − α1 β3 )E2 + − − + − + + + + − (α0 β3 + β0 α3 + α1 β2 − α2 β1 )E3 0

± ± The algebra generated by ρ , ε1 is the quaternioc subalgebra. Its divisible real form is − + ∗ − + ∗ obtained by setting: α0 = (α0 ) and α1 = (α1 ) and it is a real linear span of 1, u7, u4, u1. We now show that it is impossible to have e sextonion real algebra that has divisible quaternions − + ∗ − + ∗ as a subalgebra. To this aim we suppose α0 = (α0 ) and α1 = (α1 ) and take in (A.2) + + + − − − α0 = β0 = β1 = 0 - which implies α0 = β0 = β1 = 0. The product (A.2) shows that + − + the coefficients of ε2 and ε3 must be complex, hence each coefficient, say α2 contains 2 real − parameters a and b, and, in order to have a six dimensional real algebra, α3 viewed in R2 must be a linear transformation T of (a, b), linearity being enforced by the linearity of the algebra. + − 2 We loosely write α2 = T α3 . It is easy to show that T = Id, namely T is an involution. By + 2 + + playing with the coefficients in (A.2) we can easily obtain α2 = −T α2 = −α2 and similarly − + − for α3 , a contraddiction unless α2 = α3 = 0. This ends our proof.

Acknowledgements

We would like to thank Leron Borsten and Bruce Westbury for useful correspondence.

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