CONSTRUCTIONS FOR OCTONION AND EXCEPTIONAL JORDAN ALGEBRAS

L. J. RYLANDS AND D. E. TAYLOR

Abstract. In this note we reverse the usual process of constructing the Lie algebras of types G_{2} and F_{4} as algebras of derivations of the split octonions or the exceptional Jordan algebra and instead begin with their Dynkin diagrams and then construct the algebras together with an action of the Lie algebras and associated Chevalley groups. This is shown to be a variation on a general construction of all standard modules for simple Lie algebras and it is well suited for use in computational algebra systems. All the structure constants which occur are integral and hence the construction specialises to all fields, without restriction on the characteristic, avoiding the usual problems with characteristics 2 and 3.

1. Introduction

Traditionally, the Lie algebras of types G_{2} and F_{4} have been constructed as the algebra of derivations of the Cayley-Dickson (octonion) algebra and the exceptional Jordan algebra, respectively (see [9] and [3]). In this note it is shown that from an algorithmic and computational point of view there is an advantage in reversing the process and beginning with the Dynkin diagram defining the Lie algebra. As a result we have a uniform construction of the algebras over arbitrary fields together with explicit actions of the Lie algebra and associated Chevalley groups. For a history of other approaches see [10, Chapters VIII and IX] and the references given there, particularly [6].

2. Standard modules for semisimple Lie algebras

Where possible we choose our notation to be consistent with Humphreys [8]. Let L denote a complex semisimple Lie algebra with root system Φ, simple roots Δ, Cartan subalgebra H and Cartan decomposition

L = H ⊕ ⨁_{α∈Φ} L_{α}.

For each α ∈ Φ choose e_{α} ∈ L_{α} so that { e_{α} | α ∈ Φ } together with the elements h_{α} = [e_{α} e_{−α}] for α ∈ Δ forms a Chevalley basis for L.

Theorem 2.1. Given a subset Δ_{1} of Δ, let Φ_{1} be the root system generated by Δ_{1} and let L_{1} be the corresponding Lie subalgebra of L. For γ ∈ Φ \ Φ_{1}, let

X = { α ∈ Φ | α − γ is in the span of Δ_{1} }.

Then the subspace V of L spanned by the set {e_{α} | α ∈ X} is an L_{1}-module of dimension |X|.

Proof. The restriction of the adjoint action of L to L_{1} allows us to view L as an L_{1}-module. For α ∈ Φ_{1} and β ∈ X it is clear that α + β − γ is a linear combination of elements of Δ_{1}. Therefore, if α + β ∈ Φ, then α + β ∈ X and hence ad(e_{α})e_{β} ∈ V. To complete the proof that V is an L_{1}-submodule of L we note that we cannot have α + β = 0, otherwise −γ would belong to the span of Δ_{1}, contrary to the choice of γ.

Date: 6 January 2000.

1991 Mathematics Subject Classification. 17B25,17C40,17D05.

The standard (lowest dimensional) representation of every simple Lie algebra of type A_{n}, B_{n}, C_{n}, D_{n} or E_{n}, except that of type E_{8} (for which the standard module is the adjoint representation), can be obtained by applying this theorem to an embedding in an algebra of the same type but of rank n + 1. This observation leads to a direct and uniform means of identifying the simple Lie algebras of type A_{n}, B_{n}, C_{n} and D_{n} with the matrix algebras 𝔰𝔩(n + 1, ℂ), 𝔬(2n + 1, ℂ), 𝔰𝔭(2n, ℂ) and 𝔬(2n, ℂ). It is our purpose to describe an extension of this theorem which, in types F_{4} and G_{2}, leads to algebra structures on the modules.

The practical application of this method depends on two basic algorithms: (1) the construction of a root system from a Dynkin diagram (or Cartan matrix), and (2) the computation of structure constants for the Chevalley basis. For the root system we use the algorithm given in Humphreys [8, p. 56]. Given an ordered set of simple roots Δ = {α_{1}, …, α_{n}} this algorithm produces an ordered set Φ^{+} of positive roots in which the roots of height i appear before the roots of height j whenever i < j. For each α ∈ Φ^{+} \ Δ, let i be the least index such that α − α_{i} is a root. We choose the structure constant for [e_{α_{i}} e_{α}] to be positive and then all other structure constants are uniquely determined by the algorithm given in Carter [2, §4.2]. (For further details see [7].)

3. The octonions

Let Φ be the root system of type E_{6} and let Δ = {a, b, c, d, e, f} be a set of simple roots arranged according to the Dynkin diagram

• • • • • •_{a}  _{c}  ^{b}_{d}  _{e}  _{f}

The octonions will be constructed from the root system inclusion D_{4} ⊂ E_{6}, where D_{4} is the subsystem spanned by b, c, d and e. Let X_{ij} denote the set of roots in which a occurs with coefficient i and f occurs with coefficient j. The sets X_{11}, X_{10} and X_{01} each have 8 elements and they partition the set of 24 positive roots of E_{6} not in the D_{4} subsystem.

Let 𝔡_{4} be the Lie algebra generated by the root vectors e_{±α}, α ∈ {b, c, d, e}. Then by Theorem 2.1 the vectors e_{α} for α ∈ X_{ij} span a 𝔡_{4}-module V_{ij} and the Lie bracket induces a 𝔡_{4}-invariant bilinear pairing V_{10} × V_{01} → V_{11}. The entries in the matrices representing the action of ad(e_{α}) on V_{ij} with respect to the basis indexed by X_{ij} are the structure constants for [e_{α} e_{β}], β ∈ X_{ij}. The weights of the module V_{ij} are just the restrictions of the roots α ∈ X_{ij} to the Cartan subalgebra of 𝔡_{4} spanned by h_{b}, h_{c}, h_{d} and h_{e}. Furthermore, if we identify α ∈ X_{ij} with its row vector of coefficients and if C is the Cartan matrix of Δ, then αC gives the coefficients of α as a sum of fundamental weights. From this we see that the highest weights of the 𝔡_{4}-modules V_{11}, V_{10} and V_{01} have multiplicity one and correspond to the end nodes of the D_{4} Dynkin diagram. In particular, the modules are irreducible and pairwise non-isomorphic.

On the other hand, these modules are related by triality automorphisms of 𝔡_{4}. In general, if π is a permutation of the simple roots of 𝔡_{4} we may extend π to an automorphism of the D_{4} root system and then to an automorphism of 𝔡_{4} itself such that π(h_{α}) = h_{π(α)} and π(e_{α}) = ±e_{π(α)} for all roots α of 𝔡_{4}. In particular, let σ and τ denote the automorphisms induced by the permutations (b, c) and (b, e).

From a 𝔡_{4}-module V and an automorphism π we obtain a new 𝔡_{4}-module V^{π} by declaring the action of x ∈ 𝔡_{4} on v ∈ V to be π(x)v. By considering highest weights we see that there are isomorphisms φ : V_{11}^{σ} → V_{10} and ψ : V_{11}^{τ} → V_{01}. Similarly, the vectors e_{−α} for α ∈ X_{11} span a 𝔡_{4}-module V_{11}^{−} and we have an isomorphism μ : V_{11} → V_{11}^{−}. As all the modules V_{ij} are irreducible, these isomorphisms are unique (up to a scalar multiple).

For computational purposes we require explicit isomorphisms. To obtain these we use the fact that each module is spanned by the images of a maximal vector under the 𝔡_{4}-action ([8, §20.2]). In order to describe this we shall write the elements of Φ in an abbreviated form. For example, a + b + c + 2d + e will be written abc2de and −a − c − d will be written −acd.

As maximal vectors for V_{11}, V_{10}, V_{01} and V_{11}^{−} we take e_{a2b2c3d2ef}, e_{ab2c2de}, e_{bc2d2ef} and e_{−acdef}, respectively. These vectors are unique (up to a scalar multiple). The non-zero structure constants for the actions of e_{b}, e_{c}, e_{d} and e_{e} on the elements of X_{ij} are all 1 except that [e_{c} e_{a}] = −e_{ac}, [e_{d} e_{ac}] = −e_{acd}, [e_{e} e_{acd}] = −e_{acde} and [e_{e} e_{abcd}] = −e_{abcde}. Thus the effect of the isomorphisms φ, ψ and μ on a basis for V_{11} is given in the following table.

\begin{matrix}w ∈ V_{11} & φ(w) ∈ V_{10} & ψ(w) ∈ V_{01} & μ(w) ∈ V_{11}^{−} \\ w_{1} = e_{a2b2c3d2ef} & e_{ab2c2de} & e_{bc2d2ef} & e_{−acdef} \\ w_{2} = e_{ab2c3d2ef} & e_{abc2de} & e_{bc2def} & −e_{−abcdef} \\ w_{3} = e_{ab2c2d2ef} & e_{abcde} & e_{bcdef} & e_{−abc2def} \\ w_{4} = e_{abc2d2ef} & e_{acde} & e_{bdef} & −e_{−ab2c2def} \\ w_{5} = e_{ab2c2def} & −e_{abcd} & e_{cdef} & −e_{−abc2d2ef} \\ w_{6} = e_{abc2def} & −e_{acd} & e_{def} & e_{−ab2c2d2ef} \\ w_{7} = e_{abcdef} & e_{ac} & e_{ef} & −e_{−ab2c3d2ef} \\ w_{8} = e_{acdef} & −e_{a} & e_{f} & e_{−a2b2c3d2ef} \end{matrix}

From the isomorphism μ : V_{11} → V_{11}^{−} and the Killing form κ for 𝔢_{6} we obtain a bilinear form

(u, v) = \frac{1}{24}κ(u, μ(v))

on V_{11} which is preserved by the 𝔡_{4}-action; that is, (xu, v) + (u, xv) = 0 for all x ∈ 𝔡_{4}.

Theorem 3.1. 𝔡_{4} ≃ 𝔬(8, ℂ).

Proof. In general, the root spaces L_{α} and L_{β} are orthogonal with respect to κ if and only if α+β ≠ 0. Thus for α, β ∈ X_{11} we have (e_{α}, e_{β}) ≠ 0 if and only if α+β = 2a+2b+3c+4d+3e+2f. Then a direct calculation shows that the form (−, −) is symmetric and that the only non-zero values of (w_{i}, w_{j}) are (w_{i}, w_{9−i}) = (w_{9−i}, w_{i}) = (−1)^{i+1}, for 1 ≤ i ≤ 4. (It suffices to determine (w_{1}, w_{8}) and then to use the 𝔡_{4}-action to calculate the other values.) Thus the matrices representing 𝔡_{4} on V_{11} belong to 𝔬(8, ℂ) and on comparing dimensions we see that we have identified 𝔡_{4} with the matrix algebra 𝔬(8, ℂ).

Using the isomorphisms φ and ψ we define a bilinear product on V_{11} by

uv = [φ(u) ψ(v)].

In anticipation of its identification with the split octonions we shall denote the algebra V_{11} by 𝕆. The table of products of basis elements is readily obtained from the description of φ and ψ.

w_{1} w_{2} w_{3} w_{4} w_{5} w_{6} w_{7} w_{8}
w_{1} 0  0  0 w_{1} 0  w_{2} w_{3} w_{5}
w_{2} 0  0 −w_{1} 0 −w_{2} 0  w_{4} w_{6}
w_{3} 0  w_{1}  0 0 −w_{3} −w_{4} 0 w_{7}
w_{4} 0  w_{2} w_{3} w_{4} 0  0  0 w_{8}
w_{5} −w_{1} 0  0 0 −w_{5} −w_{6} −w_{7} 0
w_{6} −w_{2} 0  w_{5} w_{6} 0  0 −w_{8} 0
w_{7} −w_{3} −w_{5} 0 w_{7} 0  w_{8}  0 0
w_{8} −w_{4} −w_{6} −w_{7} 0 −w_{8} 0  0 0

An immediate consequence of the definition of the product is the so-called Principle of Local Triality:

x(uv) = (σ(x)u)v + u(τ(x)v) for all x ∈ 𝔡_{4}.

For α ∈ X_{ij} define the degree of e_{α} to be deg(e_{α}) = ht(α) − 8, where ht(α) is the sum of the coefficients of α with respect to the basis Δ. Let A_{i} be the subspace of 𝕆 spanned by the elements of X_{11} of degree i. Then 𝕆 = ⨁_{j=−3}^{3} A_{i} is a graded algebra.

For u = ∑_{i=1}^{8} c_{i}w_{i} ∈ 𝕆, define the trace of u to be tr(u) = c_{4} − c_{5} and the conjugate of u to be \overline{u} = tr(u)ι − u, where ι = w_{4} − w_{5}. It is clear from the multiplication table that ι is the identity element of 𝕆.

The polar form of the quadratic form n(u) = c_{1}c_{8} −c_{2}c_{7} +c_{3}c_{6} −c_{4}c_{5} is n(u+v)−n(u)−n(v) = (u, v). If 𝕆(ℤ) denotes the ℤ-span of the basis w_{1}, w_{2}, …, w_{8}, then n takes integer values on 𝕆(ℤ) and it induces a non-degenerate quadratic form on 𝕆(F) = 𝕆(ℤ) ⊗_{ℤ} F for all fields F. If the characteristic of F is not 2, then n(u) = \frac{1}{2}(u, u). We call n the norm form of 𝕆.

Lemma 3.2. For all u, v ∈ 𝕆 we have \overline{\overline{u}} = u, \overline{uv} = \overline{v} \overline{u}, n(u)ι = u \overline{u} and n(uv) = n(u)n(v).

Proof. The orthogonal complement of ι with respect to the bilinear form introduced above is spanned by w_{1}, w_{2}, w_{3}, w_{4} + w_{5}, w_{6}, w_{7} and w_{8}. The multiplication table for these elements is a skew matrix except for the elements on the secondary diagonal. There are two consequences of this observation. Firstly, that the form can be written (u, v)ι = u\overline{v} + v\overline{u} and secondly that \overline{uv} = \overline{v} \overline{u} for all u, v ∈ 𝕆. From the definitions we have tr(\overline{u}) = tr(u) and hence \overline{\overline{u}} = u. Finally, the multiplicative property of n can be checked directly from the multiplication table of the basis elements.

Theorem 3.3. The algebra 𝕆 is alternative, not associative and isomorphic to the split octonion algebra over ℂ.

Proof. This result is a consequence of the preceding lemma and the Theorem of Hurwitz [12, Theorem 3.25] but it can be established directly as follows.

Let M be the subalgebra of 𝕆 spanned by w_{1}, w_{4}, w_{5} and w_{8}. It is easily seen that M is associative and that the map given by w_{1} ↦ ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ), w_{4} ↦ ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ), w_{5} ↦ ( \begin{matrix}−1 & 0 \\ 0 & 0 \end{matrix} ) and w_{8} ↦ ( \begin{matrix}0 & 0 \\ −1 & 0 \end{matrix} ) identifies M with the algebra of 2 × 2 matrices over ℂ. The element p = w_{2} − w_{3} + w_{6} + w_{7} is orthogonal to M and therefore pu = \overline{u}p for all u ∈ M. We have p^{2} = 2ι and every element of 𝕆 can be written in the form u + vp for some u, v ∈ M. Furthermore, direct calculation with a basis for M shows that for x, y, u, v ∈ M we have (x + yp)(u + vp) = (ux + 2\overline{y}v) + (yu + v\overline{x})p and thus 𝕆 arises from M via the Cayley-Dickson process. Thus 𝕆 is uniquely determined as the split octonion algebra over ℂ (cf. [5, Section 3]).

Of course there are many ways to establish this result. In a computational context such as Lie [11], Magma [1] or GAP [13] it is a simple matter to establish that 𝕆 is alternative by checking the linearised form of the alternative law on a basis. Another approach is to begin with a vector space V of dimension three over ℂ with symmetric inner product (u, v), orthonormal basis v_{1}, v_{2}, v_{3} and the usual cross product u × v, then use Zorn’s construction for 𝕆 as the algebra of matrices ( \begin{matrix}a & v \\ w & b \end{matrix}), with a, b ∈ ℂ, v, w ∈ V and product

(\begin{matrix}a & v \\ w & b \end{matrix}) (\begin{matrix}a′ & v′ \\ w′ & b′ \end{matrix}) = (\begin{matrix}aa′ − (v, w′) & av′ + b′v + w × w′ \\ a′w + bw′ + v × v′ & bb′ − (w, v′) \end{matrix}) .

The multiplication table given in Humphreys [8, §19.3] shows that the map defined by

w_{1} ↦ (\begin{matrix}0 & 0 \\ −v_{3} & 0 \end{matrix}) , w_{2} ↦ (\begin{matrix}0 & v_{1} \\ 0 & 0 \end{matrix}) , w_{3} ↦ (\begin{matrix}0 & v_{2} \\ 0 & 0 \end{matrix}) , w_{4} ↦ (\begin{matrix}1 & 0 \\ 0 & 0 \end{matrix}) ,

w_{5} ↦ (\begin{matrix}0 & 0 \\ 0 & −1 \end{matrix}) , w_{6} ↦ (\begin{matrix}0 & 0 \\ v_{2} & 0 \end{matrix}) , w_{7} ↦ (\begin{matrix}0 & 0 \\ −v_{1} & 0 \end{matrix}) , w_{8} ↦ (\begin{matrix}0 & −v_{3} \\ 0 & 0 \end{matrix})

identifies 𝕆 with Zorn’s algebra.

It is equally easy to obtain the multiplication table given in Coxeter [4, page 561]. To this end, let e_{1} = w_{1} +w_{8}, e_{2} = w_{2} −w_{7}, e_{3} = i(w_{1} −w_{8}), e_{4} = −w_{3} −w_{6}, e_{5} = i(w_{3} −w_{6}), e_{6} = −i(w_{2} +w_{7}) and e_{7} = −i(w_{4} + w_{5}). Each of these elements squares to −ι and together with ι they form an orthonormal basis for 𝕆; hence e_{i} e_{j} = −e_{j} e_{i} for i ≠ j. Their ℝ-span is a division algebra and a real form for 𝕆. Coxeter’s relations are: e_{r} e_{s} = e_{t} for the triples (r, s, t) = (1, 2, 4), (2, 3, 5), (3, 4, 6), (4, 5, 7), (5, 6, 1), (6, 7, 2) and (7, 1, 3).

The Lie algebra 𝔤_{2} = 𝔤_{2}(ℂ) of type G_{2} may be regarded as the subalgebra of 𝔡_{4} generated by e_{A} = e_{b} + e_{c} + e_{e}, e_{B} = e_{d}, e_{−A} = e_{−b} + e_{−c} + e_{−e} and e_{−B} = e_{−d}. Its Dynkin diagram is [figure: <; A; B]. As a Chevalley basis for 𝔤_{2} we take the fixed points of the graph automorphisms σ and τ on a Chevalley basis for 𝔡_{4}. As 𝔤_{2}-modules the spaces V_{10}, V_{01} and V_{11} are isomorphic but no longer irreducible. We have xι = 0 for all x ∈ 𝔤_{2} and 𝔤_{2} acts irreducibly on the 7-dimensional space 𝕆_{0} = ι^{⊥} of elements whose trace is 0. The Principle of Local Triality shows that every element of 𝔤_{2} acts on 𝕆 as a derivation.

The elements of the Chevalley basis for 𝔤_{2} preserve the ℤ-form 𝕆(ℤ) spanned by the w_{i}. Thus our construction of 𝕆 and the associated action of 𝔤_{2} may be tensored with an arbitrary field F to give an action of 𝔤_{2}(F) on the split octonions 𝕆(F) over F.

Theorem 3.4. 𝔤_{2}(F) is the algebra of derivations of 𝕆(F) for all fields F.

Proof. For all fields, except those of characteristic 2 or 3, the argument of [3] shows that 𝔤_{2}(F) is the algebra of derivations of 𝕆(F). In characteristic 3, all maps u ↦ su − us, for s ∈ 𝕆_{0}(F) are derivations and the proof breaks down. However, the condition that a matrix be a derivation translates to a collection of linear equations in the matrix coefficients and we shall see that the space of solutions has dimension 14 in all characteristics.

We shall use the same notation (and the same multiplication table) for the basis of 𝕆(F) as we have been using for 𝕆. Then w_{1}, w_{6}, w_{7} is a basis for the subspace A_{1} = w_{5}𝕆(F)w_{4} and w_{2}, w_{3}, w_{8} is a basis for A_{2} = w_{4}𝕆(F)w_{5}. (Even though 𝕆 is not associative, A_{1} and A_{2} are well-defined.)

If D is a derivation of 𝕆(F), then D(ι) = 0 and hence D(w_{4}) = D(w_{5}). If D(w_{4}) = 0, then both A_{1} and A_{2} are D-invariant. The description given above of the isomorphism between 𝕆 and Zorn’s algebra based on the vector space V defines isomorphisms φ_{i} : A_{i} → V = F^{3} such that φ_{2}(a_{1}b_{1}) = φ_{1}(a_{1}) × φ_{1}(b_{1}) for a_{1}, b_{1} ∈ A_{1} and φ_{1}(a_{2}b_{2}) = φ_{2}(a_{2}) × φ_{2}(b_{2}) for a_{2}, b_{2} ∈ A_{2}. The restrictions of D to A_{1} and A_{2} induce linear transformations D_{1} and D_{2} of V such that D_{i}φ_{i}(a) = φ_{i}(D(a)) for all a ∈ A_{i}. It follows that D_{2}(u × v) = D_{1}(u) × v + u × D_{1}(v) and D_{1}(u × v) = D_{2}(u) × v + u × D_{2}(v). This implies that the trace of D_{1} is 0 and that D = 0 if and only if D_{1} = 0. (The matrix of D_{1} with respect to the basis v_{1}, v_{2}, v_{3} is the negative transpose of the matrix of D_{2}.) Thus the dimension of the space of derivations D such that D(w_{4}) = 0 is at most 8.

Suppose that D is an arbitrary derivation and write D(w_{4}) = aw_{4} + w where tr(w) = 0. On applying D to w_{4}w_{5} = 0 and using the fact that \overline{w}_{4} = −w_{5} we find that aw_{4} − w \overline{w}_{4} − w_{4} \overline{w} = 0. Hence a = 0 and D(w_{4}) is orthogonal to w_{4} and to w_{5}. Thus the dimension of the image of the map D ↦ D(w_{4}) is at most 6 and the dimension of its kernel is at most 8. On the other hand we already have a space of derivations 𝔤_{2}(F) of dimension 14 and this completes the proof.

4. The exceptional Jordan algebra

A Jordan algebra is a nonassociative algebra that is commutative and satisfies the Jordan identity

a^{2}(ba) = (a^{2}b)a.

It is well known (see [5, 3.11]) that the exceptional Jordan algebras are associated with the Lie algebras of type F_{4}. In this section it is shown that the exceptional Jordan algebra over ℂ can be constructed in an analogous way to the octonions in the previous section.

We begin with the root system inclusion E_{6} ⊂ E_{8}. In this case, let Φ be the root system of type E_{8}, let Δ = {a, b, c, d, e, f, g, h} be a set of simple roots arranged according to the Dynkin diagram

• • • • • • • •_{a}  _{c}  ^{b}_{d}  _{e}  _{f}  _{g}  _{h}

and let 𝔢_{6} be the Lie algebra generated by the root vectors e_{±α}, α ∈ {a, b, c, d, e, f}.

Let X_{ij} denote the set of roots in Φ which g occurs with coefficient i and h occurs with coefficient j. The sets X_{10}, X_{11} and X_{21} each have 27 elements and by Theorem 2.1, the vectors e_{α} for α ∈ X_{ij} span an 𝔢_{6}-module V_{ij}. The Lie bracket induces a bilinear pairing V_{10} × V_{11} → V_{21}, which ultimately will lead to a Jordan algebra structure on V_{21}.

The fundamental weights corresponding to the end nodes of the E_{6} sub-diagram are λ_{1} = \frac{1}{3}(4a + 3b + 5c + 6d + 4e + 2f) and λ_{6} = \frac{1}{3}(2a + 3b + 4c + 6d + 5e + 4f). Given a weight λ we follow the notation of Humphreys [8, §20.3] and use V (λ) to denote the irreducible module with highest weight λ. As maximal vectors for the modules V_{10}, V_{11} and V_{21} we have e_{2a2b3c4d3e2fg}, e_{2a2b3c4d3e2fgh} and e_{2a3b4c6d5e4f2gh}, respectively. Thus, as in the previous section, consideration of highest weights shows that the three modules are irreducible and that V_{10} and V_{11} are isomorphic to V (λ_{1}) whereas V_{21} is isomorphic to V (λ_{6}). Note that ad(e_{h}) : V_{10} → V_{11} is an 𝔢_{6}-isomorphism.

Instead of triality we now have a duality automorphism of 𝔢_{6} induced by the permutation π = (a, f)(c, e) of the simple roots. This automorphism interchanges the weights λ_{1} and λ_{6} and we have 𝔢_{6}-isomorphisms φ : V_{21}^{π} → V_{10} and ψ : V_{21}^{π} → V_{11}, where V_{21}^{π} is the vector space V_{21} with the action of x ∈ 𝔢_{6} on v ∈ V_{21} given by π(x)v. Similarly, the vectors e_{−α} with α ∈ X_{21} span an 𝔢_{6}-module V_{21}^{−} with maximal vector e_{−bc2d2e2f2gh} and highest weight λ_{1}. Hence there is an isomorphism μ : V_{21}^{π} → V_{21}^{−}. The Killing form κ for 𝔢_{8} induces a non-degenerate pairing V_{21}^{−} × V_{21} → ℂ and therefore V_{21}^{−} is the dual of V_{21}.

Providing a table to describe the isomorphisms φ, ψ and μ as we did for the octonions is too cumbersome and instead we simply exhibit a sequence of 26 pairs (i_{k}, α_{k}) where 1 ≤ i_{k} ≤ 26 and α_{k} ∈ {a, b, c, d, e, f}. We shall use these to produce compatible bases for V_{21}^{π}, V_{10} and V_{11}.

(1, a), (2, c), (3, d), (4, b), (4, e), (6, f), (6, b), (7, b), (8, d),
(9, d), (10, c), (11, e), (11, c), (12, a), (13, c), (14, a), (16, d), (16, a),
(18, b), (18, a), (20, a), (21, c), (22, c), (24, d), (25, e), (26, f).

An ordered basis z_{1}, z_{2}, …, z_{27} for V_{21}^{π} can be obtained by taking z_{1} to be the maximal vector in X_{21} and recursively setting z_{k+1} = e_{−α_{k}} · z_{k} = ad(e_{−π(α_{k})})z_{i_{k}}, where (i_{k}, α_{k}) is the k-th pair from the above list. Given the basis { z_{k} | 1 ≤ k ≤ 27 } for V_{21}^{π}, the basis { φ(z_{k}) | 1 ≤ k ≤ 27 } for V_{10} is obtained by taking φ(z_{1}) to be the maximal vector in X_{10} and carrying out the process described in the previous sentence. (Similarly we take ψ(z_{1}) to be the maximal vector in X_{11} and apply the algorithm to obtain ψ(z_{k}).) Our choice of structure constants means that the matrices representing ad(e_{a}), …, ad(e_{f}) with respect to any of the bases indexed by X_{10}, X_{11} or X_{21} have only six non-zero entries, all of which are 1, and the bases obtained here are permutations of the original bases.

On the other hand, the matrices representing ad(e_{a}), …, ad(e_{f}) on V_{21}^{−} are the negatives of the matrices representing the action on V_{10} and so the new basis for V_{21}^{−} is a signed permutation of the original basis. In fact, the sign attached to e_{−α} is (−1)^{deg(α)}, where in this case deg(α) = ht(α) − 19.

The obvious multiplication on V_{21} defined by u ∘ v = [φ(u) ψ(v)] is commutative and for x ∈ 𝔢_{6} we have a “Principle of Local Duality”:

x(u ∘ v) = π(x)u ∘ v + u ∘ π(x)v.

But the algebra defined by this multiplication does not have an identity element, nor does it satisfy the Jordan identity. Therefore we modify our definition to the following:

u ⋆ v = [φ(u) ψ(v)] + \frac{1}{60}κ(u, μ(v))ι,

where κ is the Killing form for 𝔢_{8} and ι = z_{13} − z_{14} + z_{15}. (A direct calculation shows that κ(e_{α}, e_{−α}) = 60 for all α ∈ X_{21}.)

From now on we use 𝕁 to denote the algebra (V_{21}, ⋆). If B_{i} is the subspace of 𝕁 spanned by the elements of X_{21} of degree i, then 𝕁 = ⨁_{j=−8}^{8} B_{i} exhibits 𝕁 as a graded algebra.

The well-known 𝔢_{6}-invariant cubic form on V_{21}^{π} can be defined by

τ(u, v, w) = κ([φ(u) ψ(v)], μ(w)).

The Lie algebra 𝔣_{4} may be regarded as the subalgebra of 𝔢_{6} generated by e_{A} = e_{a} + e_{f}, e_{B} = e_{c} + e_{e}, e_{C} = e_{d}, e_{D} = e_{b}, e_{−A} = e_{−a} + e_{−f}, e_{−B} = e_{−c} + e_{−e}, e_{−C} = e_{−d} and e_{−D} = e_{−b}. These elements are fixed by the graph automorphism π and their Dynkin diagram is • • • •^{<}_{A} _{B} _{C} _{D}.

As 𝔣_{4}-modules the spaces V_{10}, V_{11} and V_{21} are isomorphic but no longer irreducible.

The 𝔣_{4}-invariant bilinear form (u, v) = \frac{1}{60}κ(u, μ(v)) on 𝕁 is symmetric and takes only the values 0 and ±1 on the basis {z_{k}}. On the other hand, (ι, ι) = 3 and so the splitting 𝕁 = 𝕁_{0} ⊕ ⟨ ι ⟩, where 𝕁_{0} is the orthogonal complement of ι, persists over all fields except those of characteristic 3. The elements of 𝔣_{4} act as derivations of 𝕁.

Theorem 4.1. The algebra 𝕁 is the exceptional Jordan algebra over ℂ; its identity element is ι.

Proof. We shall identify 𝕁 with the algebra ℌ(3, 𝕆) of 3×3 hermitian matrices over 𝕆 described in Schafer [12, Chap. IV §2] except that we define the multiplication in ℌ(3, 𝕆) to be x·y = xy +yx. This has the effect of making \frac{1}{2}I the identity element (where I is the identity matrix) but it simplifies the description of the isomorphism.

The elements of ℌ(3, 𝕆) have the form

Ξ(ξ, η, θ; a, b, c) = (\begin{matrix}ξ & c & \overline{b} \\ \overline{c} & η & a \\ b & \overline{a} & θ \end{matrix})

where ξ, η, θ ∈ ℂ and a, b, c ∈ 𝕆.

If we put e_{1} = z_{13}−z_{14}, e_{2} = z_{13}+z_{15} and e_{3} = −z_{14}+z_{15}, then the elements \frac{1}{2}e_{i} are orthogonal idempotents which sum to ι. The subspaces A = e_{2} ⋆ (𝕁 ⋆ e_{3}), B = e_{3} ⋆ (𝕁 ⋆ e_{1}) and C = e_{1} ⋆ (𝕁 ⋆ e_{2}) are of dimension 8 and we shall identify their underlying vector spaces with 𝕆. For example, we may take the basis z_{1}, z_{7}, z_{9}, z_{11}, z_{16}, z_{18}, z_{20}, z_{27} for A to correspond to the basis w_{1}, w_{2}, …, w_{8} for 𝕆 obtained in Section 3. Similar identifications (with minor variations) apply to B and C. In fact it is a straightforward matter to use the multiplication tables given for the w_{i} and z_{j} to

\begin{matrix}⋆ & z_{1} & z_{2} & z_{3} & z_{4} & z_{5} & z_{6} & z_{7} & z_{8} & z_{9} & z_{10} & z_{11} & z_{12} & z_{13} & z_{14} & z_{15} & z_{16} & z_{17} & z_{18} & z_{19} & z_{20} & z_{21} & z_{22} & z_{23} & z_{24} & z_{25} & z_{26} & z_{27} \\ z_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & z_{1} & 0 & z_{2} & 0 & z_{3} & 0 & z_{4} & z_{5} & z_{6} & z_{8} & z_{10} & z_{12} & z_{15}+ι \\ z_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & −z_{1} & 0 & −z_{2} & 0 & −z_{3} & 0 & −z_{4} & 0 & −z_{5} & 0 & 0 & z_{7} & z_{9} & z_{11} & z_{14}−ι & z_{17} \\ z_{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & z_{1} & z_{2} & 0 & z_{3} & 0 & 0 & 0 & 0 & −z_{6} & 0 & −z_{8} & −z_{7} & −z_{9} & 0 & 0 & z_{13}+ι & z_{16} & z_{19} \\ z_{4} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & −z_{1} & −z_{2} & 0 & 0 & 0 & z_{4} & 0 & 0 & z_{6} & 0 & 0 & z_{7} & −z_{10} & 0 & −z_{11} & 0 & −z_{13}−ι & 0 & z_{18} & z_{21} \\ z_{5} & 0 & 0 & 0 & 0 & 0 & z_{1} & z_{2} & 0 & 0 & 0 & 0 & 0 & z_{5} & 0 & 0 & z_{8} & 0 & z_{10} & z_{9} & 0 & z_{11} & 0 & z_{13}+ι & 0 & 0 & z_{20} & z_{22} \\ z_{6} & 0 & 0 & 0 & 0 & z_{1} & 0 & 0 & 0 & −z_{3} & 0 & −z_{4} & 0 & 0 & −z_{6} & 0 & 0 & −z_{7} & 0 & 0 & −z_{12} & 0 & −z_{14}+ι & 0 & −z_{16} & −z_{18} & 0 & z_{23} \\ z_{7} & 0 & 0 & 0 & 0 & z_{2} & 0 & 0 & z_{3} & 0 & z_{4} & 0 & z_{6} & 0 & 0 & z_{7} & 0 & 0 & 0 & 0 & −z_{15}−ι & 0 & −z_{17} & 0 & −z_{19} & −z_{21} & −z_{23} & 0 \\ z_{8} & 0 & 0 & 0 & −z_{1} & 0 & 0 & z_{3} & 0 & 0 & 0 & −z_{5} & 0 & 0 & −z_{8} & 0 & 0 & −z_{9} & z_{12} & 0 & 0 & z_{14}−ι & 0 & z_{16} & 0 & −z_{20} & 0 & z_{24} \\ z_{9} & 0 & 0 & 0 & −z_{2} & 0 & −z_{3} & 0 & 0 & 0 & z_{5} & 0 & z_{8} & 0 & 0 & z_{9} & 0 & 0 & z_{15}+ι & 0 & 0 & z_{17} & 0 & z_{19} & 0 & −z_{22} & −z_{24} & 0 \\ z_{10} & 0 & 0 & z_{1} & 0 & 0 & 0 & z_{4} & 0 & z_{5} & 0 & 0 & 0 & 0 & −z_{10} & 0 & −z_{12} & −z_{11} & 0 & −z_{14}+ι & 0 & 0 & 0 & z_{18} & z_{20} & 0 & 0 & z_{25} \\ z_{11} & 0 & 0 & z_{2} & 0 & 0 & −z_{4} & 0 & −z_{5} & 0 & 0 & 0 & z_{10} & 0 & 0 & z_{11} & −z_{15}−ι & 0 & 0 & −z_{17} & 0 & 0 & 0 & z_{21} & z_{22} & 0 & −z_{25} & 0 \\ z_{12} & 0 & −z_{1} & 0 & 0 & 0 & 0 & z_{6} & 0 & z_{8} & 0 & z_{10} & 0 & z_{12} & 0 & 0 & 0 & −z_{13}−ι & 0 & −z_{16} & 0 & −z_{18} & −z_{20} & 0 & 0 & 0 & 0 & z_{26} \\ z_{13} & 0 & 0 & z_{3} & z_{4} & z_{5} & 0 & 0 & 0 & 0 & 0 & 0 & z_{12} & ι & z_{15} & z_{14} & 0 & z_{17} & 0 & 0 & 0 & 0 & 0 & z_{23} & z_{24} & z_{25} & 0 & 0 \\ z_{14} & 0 & −z_{2} & 0 & 0 & 0 & −z_{6} & 0 & −z_{8} & 0 & −z_{10} & 0 & 0 & z_{15} & ι & z_{13} & 0 & 0 & 0 & −z_{19} & 0 & −z_{21} & −z_{22} & 0 & 0 & 0 & −z_{26} & 0 \\ z_{15} & z_{1} & 0 & 0 & 0 & 0 & 0 & z_{7} & 0 & z_{9} & 0 & z_{11} & 0 & z_{14} & z_{13} & ι & z_{16} & 0 & z_{18} & 0 & z_{20} & 0 & 0 & 0 & 0 & 0 & 0 & z_{27} \\ z_{16} & 0 & −z_{3} & 0 & z_{6} & z_{8} & 0 & 0 & 0 & 0 & −z_{12} & −z_{15}−ι & 0 & 0 & 0 & z_{16} & 0 & z_{19} & 0 & 0 & 0 & −z_{23} & −z_{24} & 0 & 0 & z_{26} & 0 & 0 \\ z_{17} & z_{2} & 0 & 0 & 0 & 0 & −z_{7} & 0 & −z_{9} & 0 & −z_{11} & 0 & −z_{13}−ι & z_{17} & 0 & 0 & z_{19} & 0 & z_{21} & 0 & z_{22} & 0 & 0 & 0 & 0 & 0 & −z_{27} & 0 \\ z_{18} & 0 & −z_{4} & −z_{6} & 0 & z_{10} & 0 & 0 & z_{12} & z_{15}+ι & 0 & 0 & 0 & 0 & 0 & z_{18} & 0 & z_{21} & 0 & z_{23} & 0 & 0 & −z_{25} & 0 & −z_{26} & 0 & 0 & 0 \\ z_{19} & z_{3} & 0 & 0 & z_{7} & z_{9} & 0 & 0 & 0 & 0 & −z_{14}+ι & −z_{17} & −z_{16} & 0 & −z_{19} & 0 & 0 & 0 & z_{23} & 0 & z_{24} & 0 & 0 & 0 & 0 & z_{27} & 0 & 0 \\ z_{20} & 0 & −z_{5} & −z_{8} & −z_{10} & 0 & −z_{12} & −z_{15}−ι & 0 & 0 & 0 & 0 & 0 & 0 & 0 & z_{20} & 0 & z_{22} & 0 & z_{24} & 0 & z_{25} & 0 & z_{26} & 0 & 0 & 0 & 0 \\ z_{21} & z_{4} & 0 & −z_{7} & 0 & z_{11} & 0 & 0 & z_{14}−ι & z_{17} & 0 & 0 & −z_{18} & 0 & −z_{21} & 0 & −z_{23} & 0 & 0 & 0 & z_{25} & 0 & 0 & 0 & −z_{27} & 0 & 0 & 0 \\ z_{22} & z_{5} & 0 & −z_{9} & −z_{11} & 0 & −z_{14}+ι & −z_{17} & 0 & 0 & 0 & 0 & −z_{20} & 0 & −z_{22} & 0 & −z_{24} & 0 & −z_{25} & 0 & 0 & 0 & 0 & z_{27} & 0 & 0 & 0 & 0 \\ z_{23} & z_{6} & z_{7} & 0 & 0 & z_{13}+ι & 0 & 0 & z_{16} & z_{19} & z_{18} & z_{21} & 0 & z_{23} & 0 & 0 & 0 & 0 & 0 & 0 & z_{26} & 0 & z_{27} & 0 & 0 & 0 & 0 & 0 \\ z_{24} & z_{8} & z_{9} & 0 & −z_{13}−ι & 0 & −z_{16} & −z_{19} & 0 & 0 & z_{20} & z_{22} & 0 & z_{24} & 0 & 0 & 0 & 0 & −z_{26} & 0 & 0 & −z_{27} & 0 & 0 & 0 & 0 & 0 & 0 \\ z_{25} & z_{10} & z_{11} & z_{13}+ι & 0 & 0 & −z_{18} & −z_{21} & −z_{20} & −z_{22} & 0 & 0 & 0 & z_{25} & 0 & 0 & z_{26} & 0 & 0 & z_{27} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ z_{26} & z_{12} & z_{14}−ι & z_{16} & z_{18} & z_{20} & 0 & −z_{23} & 0 & −z_{24} & 0 & −z_{25} & 0 & 0 & −z_{26} & 0 & 0 & −z_{27} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ z_{27} & z_{15}+ι & z_{17} & z_{19} & z_{21} & z_{22} & z_{23} & 0 & z_{24} & 0 & z_{25} & 0 & z_{26} & 0 & 0 & z_{27} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}

Figure 1. Multiplication table for a basis of the exceptional Jordan algebra check that the following mapping defines an isomorphism 𝕁 → ℌ(3, 𝕆).

z_{1} ↦ Ξ(0, 0, 0 ; w_{1}, 0, 0)  z_{2} ↦ Ξ(0, 0, 0 ; 0, −w_{1}, 0)  z_{3} ↦ Ξ(0, 0, 0 ; 0, 0, w_{1})
z_{4} ↦ Ξ(0, 0, 0 ; 0, 0, w_{2})  z_{5} ↦ Ξ(0, 0, 0 ; 0, 0, w_{3})  z_{6} ↦ Ξ(0, 0, 0 ; 0, w_{2}, 0)
z_{7} ↦ Ξ(0, 0, 0 ; w_{2}, 0, 0)  z_{8} ↦ Ξ(0, 0, 0 ; 0, w_{3}, 0)  z_{9} ↦ Ξ(0, 0, 0 ; w_{3}, 0, 0)
z_{10} ↦ Ξ(0, 0, 0 ; 0, w_{5}, 0)  z_{11} ↦ Ξ(0, 0, 0 ; w_{4}, 0, 0)  z_{12} ↦ Ξ(0, 0, 0 ; 0, 0, −w_{4})
z_{13} ↦ Ξ(\frac{1}{2}, \frac{1}{2}, −\frac{1}{2} ; 0, 0, 0)  z_{14} ↦ Ξ(−\frac{1}{2}, \frac{1}{2}, −\frac{1}{2} ; 0, 0, 0)  z_{15} ↦ Ξ(−\frac{1}{2}, \frac{1}{2}, \frac{1}{2} ; 0, 0, 0)
z_{16} ↦ Ξ(0, 0, 0 ; w_{5}, 0, 0)  z_{17} ↦ Ξ(0, 0, 0 ; 0, 0, −w_{5})  z_{18} ↦ Ξ(0, 0, 0 ; w_{6}, 0, 0)
z_{19} ↦ Ξ(0, 0, 0 ; 0, −w_{4}, 0)  z_{20} ↦ Ξ(0, 0, 0 ; w_{7}, 0, 0)  z_{21} ↦ Ξ(0, 0, 0 ; 0, −w_{6}, 0)
z_{22} ↦ Ξ(0, 0, 0 ; 0, −w_{7}, 0)  z_{23} ↦ Ξ(0, 0, 0 ; 0, 0, w_{6})  z_{24} ↦ Ξ(0, 0, 0 ; 0, 0, w_{7})
z_{25} ↦ Ξ(0, 0, 0 ; 0, 0, w_{8})  z_{26} ↦ Ξ(0, 0, 0 ; 0, w_{8}, 0)  z_{27} ↦ Ξ(0, 0, 0 ; w_{8}, 0, 0)

This completes the proof.

As a result of this theorem we know that 𝕁 satisfies the Jordan identity. On the other hand, the linearised form of this identity can be written

(x, y, w ⋆ z) + (w, y, z ⋆ x) + (z, y, x ⋆ w) = 0,

where (x, y, z) = (x ⋆ y) ⋆ z − x ⋆ (y ⋆ z) is the associator (see [12, (4.3)]) and a computer algebra system such as Magma [1] can be used to check directly that this holds for the basis {z_{k}}.

We have seen that the 𝔣_{4}-module V_{21} has many multiplications preserved by 𝔣_{4}. In the next theorem we show that only one of these gives a Jordan algebra.

Theorem 4.2. Up to a scalar multiple the multiplication defining 𝕁 gives the unique Jordan algebra structure on V_{21}.

Proof. Let μ_{1}, μ_{2}, μ_{3} and μ_{4} denote the fundamental weights for F_{4}. Then μ_{1} = 2A+3B +2C +D and 𝕁_{0} ≃ V (μ_{1}). Using the formulae of Weyl and Freudenthal (Humphreys [8]) we find that the symmetric and exterior squares of V (μ_{1}) decompose as follows:

S^{2}V (μ_{1}) = V (0) ⊕ V (μ_{1}) ⊕ V (2μ_{1}) and Λ^{2}V (μ_{1}) = V (μ_{2}) ⊕ V (μ_{4}).

Thus (up to scalars) 𝔣_{4} preserves a unique multiplication and a unique bilinear form on 𝕁_{0}, both of which are symmetric. It follows that any 𝔣_{4}-invariant multiplication on 𝕁 must have the form

u ∘ v = a[φ(u) ψ(v)] + b\frac{1}{60}κ(u, μ(v))ι,

for some a and b ∈ ℂ. We may choose the scalars so that ι is the identity and then by checking the Jordan relation on a few basis elements we find that we must have a = b = 1.

Let 𝕁(ℤ) denote the ℤ-form spanned by the basis z_{1}, z_{2}, …, z_{27}. The structure constants for this basis belong to ℤ and thus for any field F the space 𝕁(F) = 𝕁(ℤ) ⊗_{ℤ} F inherits an algebra structure from 𝕁 on which 𝔣_{4}(F) acts as derivations.

Theorem 4.3. 𝔣_{4}(F) is the algebra of derivations of 𝕁(F) for all fields F.

Proof. We use the notation introduced in the proof of Theorem 4.1 but we carry out our calculations in F. In particular, let A = e_{2} ⋆ (𝕁 ⋆ e_{3}), B = e_{3} ⋆ (𝕁 ⋆ e_{1}) and C = e_{1} ⋆ (𝕁 ⋆ e_{2}).

Suppose at first that the characteristic of F is not 2 and consider the action of a derivation D on the elements e_{1} = z_{13} − z_{14}, e_{2} = z_{13} + z_{15} and e_{3} = −z_{14} + z_{15}. We have D(e_{i}) ⋆ e_{i} = D(e_{i}) for all i. Furthermore, the elements u ∈ 𝕁(F) such that u ⋆ e_{1} = u constitute the subspace B ⊕ C and for such elements u we have u ⋆ e_{2} ∈ C. Similar observations apply to e_{2} and e_{3}. It follows that the image of the map χ : D ↦ D(e_{1}) ⋆ e_{2} + D(e_{2}) ⋆ e_{3} + D(e_{3}) ⋆ e_{1} is contained in A ⊕ B ⊕ C, and thus its dimension is at most 24.

Suppose that χ(D) = 0. Then D(e_{1}) ⋆ e_{2} = D(e_{1}) ⋆ e_{3} = 0 and D(e_{1}) ⋆ e_{1} = D(e_{1}). Hence 2D(e_{1})⋆ι = D(e_{1}) and so D(e_{1}) = 0. Similarly, D(e_{2}) = D(e_{3}) = 0. From its definition, it is clear that A is preserved by D. Moreover, for all x, y ∈ A we can check from the multiplication table that x ⋆ y = (x, y)(e_{2} + e_{3}). Thus ((Dx, y) + (x, Dy))(e_{2} + e_{3}) = (Dx) ⋆ y + x ⋆ Dy = D(x ⋆ y) = 0. That is, D preserves the non-degenerate form (u, v) on A and hence the restriction of D to A belongs to 𝔡_{4}(F). The identification of 𝕁 with ℌ(3, 𝕆) used in Theorem 4.1 extends to any field of characteristic ≠ 2 and identifies A, B and C with 𝕆(F). Furthermore, as shown in [3], the restrictions of D to A, B and C are related by triality. Thus if D(a) = 0 for all a ∈ A, then D = 0. Consequently the kernel of χ has dimension at most 28 and it follows from the previous paragraph that the space of derivations has dimension at most 52. On the other hand we already have a space of derivations 𝔣_{4}(F) of dimension 52.

Now suppose that the characteristic of F is 2 and that D is a derivation. Then z_{1} ⋆ z_{27} = z_{11} ⋆ z_{16} = e_{1} and on applying D we see from the multiplication table that D(e_{1}) ⋆ e_{2} ∈ C. Similarly, D(e_{2}) ⋆ e_{3} ∈ A and D(e_{3}) ⋆ e_{1} ∈ B. Defining χ as before we again deduce that the dimension of the image of χ is at most 24.

From now on assume that χ(D) = 0. Then D(e_{1}) ⋆ e_{2} = D(e_{1}) ⋆ e_{3} = 0 and so D(e_{1}) ∈ ⟨ z_{13}, z_{14}, z_{15} ⟩. On applying D to the equality e_{1} ⋆ z_{2} = z_{2} we see that D(z_{2}) ∈ B ⊕ C and D(e_{1}) ∈ ⟨ z_{13}, z_{15} ⟩. Applying D to other equalities of the form e_{i} ⋆ z_{j} = z_{j} shows that D preserves A, B and C and that D(e_{1}) ∈ ⟨ z_{15} ⟩, D(e_{2}) ∈ ⟨ z_{14} ⟩ and D(e_{3}) ∈ ⟨ z_{13} ⟩. Further calculations of a similar nature show that D(z_{13}) = D(z_{14}) = D(z_{15}) = 0.

The bilinear form (−, −) on 𝕁 restricts to a non-degenerate symplectic form on A and, as shown above, (Dx, y)+(x, Dy) = 0 for all x, y ∈ A. That is, D|_{A} belongs to the 36-dimensional symplectic Lie algebra 𝔰𝔭(8, F). Consideration of the action of D on equalities of the form z_{1} ⋆ z_{10} = 0 shows that (Dx, x) = 0 for all x ∈ A. This means that D|_{A} ∈ 𝔬(8, F), the 28-dimensional orthogonal Lie algebra.

Consider a derivation T on 𝕁 such that χ(T) = 0, T (z_{1}) = αz_{1}, T (z_{27}) = αz_{27} and T (z_{i}) = 0 for i ∈ {7, 9, 11, 16, 18, 20}. The action of T on equalities of the form z_{i} ⋆ z_{j} = z_{k} relating elements of A, B and C shows that α = 0 and that T acts as a scalar on B ⊕C. There are two consequences of this calculation. Firstly, that not every element of 𝔬(8, F) can be written in the form D|_{A} where χ(D) = 0 and secondly that the space of derivations D such that χ(D) = 0 and which vanish on A has dimension at most 1.

This shows that the dimension of the space of derivations of 𝕁 is at most 24 + 27 + 1 = 52. As in the previous case, this proves that 𝔣_{4}(F) is the space of derivations of 𝕁.

5. Implementation

Recent versions of Magma [1], GAP [13], Maple [14] and Lie [11] have the ability to construct root systems. Magma 2.5 has support for structure constants, making it an easy task to implement these algebras and to produce the tables used in this paper.

6. Acknowledgements

Some of the results in this paper were presented to the conference on Geometric and Algebraic Combinatorics, held in Oisterwijk, The Netherlands (August 1999), in honour of J. J. Seidel on the occasion of his 80th birthday.

References

1. W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system I: The user language, J. Symbolic Comput. 24 (1997), 235–265.

2. R. W. Carter, Simple groups of Lie type, Pure and Applied Mathematics, vol. XXVIII, John Wiley & Sons, London New York Sydney Toronto, 1972.

3. C. Chevalley and R. D. Schafer, The exceptional simple Lie algebras F_{4} and E_{6}, Proc. Nat. Acad. Sci. U.S.A. 36 (1950), 137–141.

4. H. S. M. Coxeter, Integral Cayley numbers, Duke Math. J. 13 (1946), 561–578.

5. J. R. Faulkner and J. C. Ferrar, Exceptional Lie algebras and related algebraic and geometric structures, Bull. London Math. Soc. 9 (1977), 1–35.

6. H. Freudenthal, Oktaven, Ausnahmegruppen und Oktavengeometrie, Geom. Dedicata 19 (1985), 1–63.

7. R. B. Howlett, L. J. Rylands, and D. E. Taylor, Matrix generators for exceptional groups of Lie type, submitted J. Symbolic Comput. 00 (2000), 000–000.

8. J. E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, vol. 9, Springer-Verlag, New York Heidelberg Berlin, 1972.

9. N. Jacobson, Cayley numbers and simple Lie algebras of type G_{2}, Duke Math. J. 5 (1939), 775–783.

10. M.-A. Knus, A. Merkurjev, M. Rost, and J.-P. Tignol, The Book of Involutions, Colloquium Publications, vol. 44, Amer. Math. Soc., Providence, RI, 1998.

11. M. A. A. van Leeuwen, A. M. Cohen, and B. Lisser, Lie Manual, version 2.1, Computer Algebra Group, CWI, Amsterdam, 1993.

12. R. D. Schafer, An Introduction to Nonassociative Algebras, Pure and Applied Mathematics, vol. 22, Academic Press, New York London, 1966.

13. M. Schönert et al., GAP — Groups, Algorithms and Programming, Lehrstuhl D für Mathematik, RWTH Aachen, 1994.

14. J. R. Stembridge, A Maple package for root systems and finite Coxeter groups (Version 2.1), Maple Share Library, 1998.

L. J. Rylands, School of Science, University of Western Sydney, Nepean, Australia

E-mail address: L.Rylands@nepean.uws.edu.au

D. E. Taylor, School of Mathematics and Statistics, University of Sydney, NSW 2006 Australia

E-mail address: D.Taylor@maths.usyd.edu.au
URL: http://www.maths.usyd.edu.au:8000/u/don/