Title. Author(s)YAMAGUCHI, KEIZO. CitationHokkaido University Preprint Series in Mathematics, Issue Date DOI / Doc URL.

Title G2- GEOMETRY IN CONTACT GEOMETRY OF SECOND ORDER Author(s)YAMAGUCHI, KEIZO CitationHokkaido University Preprint Series in Mathematics, Issue Date 2018-05-15 DOI 10.14943/84415 Doc URL http://hdl.handle.net/2115/70227 Type bulletin (article) File Information G_2-geometry.pdf Instructions for use Hokkaido University Collection of Scholarly and Aca

G 2 - GEOMETRY IN CONTACT GEOMETRY OF SECOND ORDER KEIZO YAMAGUCHI 1. Introduction In his famous Five variables paper [3], E. Cartan investigated the (local) contact equivalence problem of two classes of second order partial differential equations for a scalar function in two independent variables. One class consists of overdetermined systems, which are involutive, and the other class consists of single equations of Goursat type, i.e., single equations of parabolic type whose Monge characteristic systems are completely integrable. Especially, in [3], he found out the following facts: the symmetry algebras (i.e., the Lie algebras of infinitesimal contact transformations) of the following overdetermined system (A) and the single Goursat type equation (B) are both isomorphic with the 14-dimensional exceptional simple Lie algebra G 2. (A) 2 z x 2 = 1 3 ( ) 2 3 z, y 2 2 z x y = 1 2 ( ) 2 2 z. y 2 where (B) 9r 2 + 12t 2 (rt s 2 ) + 32s 3 36rst = 0, r = 2 z x, s = 2 z 2 x y, t = 2 z y. 2 are the classical terminology. In [13], we observed, for each exceptional simple Lie algebra X l, we could find the overdetermined system (A l ) and the single equation of Goursat type (B l ), whose symmetry algebras are isomporphic with X l and formulated this fact as the G 2 -geometry. We will first recall this observation in 2. The main purpose of the present paper is to construct the (local) models for overdetermined systems (A l ) explicitly for each exceptional simple Lie algebra and also for the classical type analogy, which will be carried out in 3 and 4. We will also give parametric descriptions of the single equation of Goursat type (B l ) in 5. See [8], for the recent development of this subject. Throughout this paper, we follow the terminology and notations of our previous papers [12], [13], [14] and [15]. 2. G 2 -geometry 2.1. Standard Contact Manifolds. Let g be a finite dimensional simple Lie algebra over C. Let us fix a Cartan subalgebra h of g and choose a simple root system = {α 1,..., α l } of the root system Φ of g relative to h. Each simple Lie algebra g over C has the highest root θ. Let θ denote the subset of consisting of all vertices which are connected to θ in the Extended Dynkin diagram of X l (l 2). This subset θ of, by the construction in 3.3 [12], defines a gradation (or a partition of Φ + ), which distinguishes the highest root θ. Then, this gradation (X l, θ ) turns out to be a contact gradation, which is unique up to conjugacy This work was partially supported by the grant 346300 for IMPAN from the Simon Foundation and the matching 2015-2019 Polish MNiSW fund. 1

(Theorem 4.1 [12]). Explicitly we have θ = {α 1, α l } for A l type and θ = {α θ } for other types. Here α θ = α 2, α 1, α 2 for B l, C l, D l types respectively and α θ = α 2, α 1, α 2, α 1, α 8 for G 2, F 4, E 6, E 7, E 8 types respectively. Moreover we have the adjoint (or equivalently coadjoint) representation, which has θ as the highest weight. The R-space J g corresponding to (X l, θ ) can be obtained as the projectiviation of the (co-)adjoint orbit of the adjoint group G of g passing through the root vector of θ. By this construction, J g has the natural contact structure C g induced from the symplectic structure as the coadjoint orbit, which corresponds to the contact gradation (X l, θ ) (cf. [12], 4). Standard contact manifolds (J g, C g ) were first found by Boothby ([1]) as compact simply connected homogeneous complex contact manifolds. For the explicit description of the standard contact manifolds of the classical type, we refer the reader to 4.3 [12]. Extended Dynkin Diagrams with the coefficient of Highest Root (cf. [2]) θ 1 1 1 1... θ 2 2 2... 1 = α 2 α l 1 α l α 1 α 1 α 2 α l 1 α l A l (l > 1) B l (l > 2) 2 2 1 =... 2 2 = θ α 1 α l 1 α l... 1 1 α 2 α l 2 α 1 C l (l > 1) D l (l > 3) 1 2 3 2 1 2 3 4 2 α 1 α 3 α 4 α 5 α 6 = 2 θ α 1 α 2 α 3 α 4 α 2 E 6 F 4 θ 2 3 4 3 2 1 = θ α 1 α 3 α 4 α 5 α 6 α 7 α 1 α 2 θ 2 α 2 E 7 G 2 2 4 6 5 4 3 2 α 1 α 3 α 4 α 5 α 6 α 7 α 8 3 α 2 E 8 θ θ 3 2 1 α l 1 α l 2.2. G 2 -geometry. Let (X l, θ ) be the (standard) contact gradation. Then we have θ = {α θ } except for A l type (see above). As we observed in 6.3 in [13], for the exceptional simple Lie algebras, there exists, without exception, a unique simple root α G next to α θ such that the coefficient of α G in the highest root is 3 (see the above diagrams). In the classical cases, the pair {α 1, α 3 } of simple roots plays the role of α G in BD l type. We will consider simple graded Lie algebras (X l, {α G }) of depth 3 and let m = g 3 g 2 g 1 be its negative part. We will call this gradation m = g 3 g 2 g 1 the Goursat gradation of type X l. Moreover we will show that regular differential systems (X, D) of type m satisfy the conditions (X.1) to (X.3) in 4.3 [14] so that we can construct P D manifolds (R(X), DX 1, D2 X ) from (X, D). Our 2

model overdetermined system (A l ) will be the P D manifold of second order constructed from the standard differential system of type m, where m is the Goursat gradation of type X l. Explicitly we will here consider the following simple graded Lie algebras of depth 3: (G 2, {α 1 }), (F 4, {α 2 }), (E 6, {α 4 }), (E 7, {α 3 }), (E 8, {α 7 }), (B l, {α 1, α 3 }) (l 3), (D l, {α 1, α 3 }) (l 5) and (D 4, {α 1, α 3, α 4 }). These graded Lie algebras have the common feature with (G 2, {α 1 }) as follows: The Goursat gradation m = g 3 g 2 g 1 satisfies dim g 3 = 2 and dim g 1 = 2 dim g 2. In fact, in the description of the gradation in terms of the root space decomposition in 3.3 [12], in each case, we can check that Φ + 3 = {θ, θ α θ } such that the coefficient of α θ in each β Φ + 2 is 1 and Φ + 1 consists of roots θ β, θ α θ β for each β Φ + 2 (see 3 and 4 for detail). Hence, ignoring the bracket product in g 1, we can describe the bracket products of other part of m, in terms of paring, by g 3 = W, g 2 = V and g 1 = W V, where dim W = 2 and dim V = s. Here s = 1, 6, 9, 15, 27, 2l 4 or 2l 5 corrresponding to X l = G 2, F 4, E 6, E 7, E 8, B l or D l. Thus let (X, D) be a regular differential system of type m, where m is the Goursat gradation of type X l. Then (X, D) is a regular differential system of type c 1 (s, 2), where c 1 (s, 2) = g 2 g 1, g 2 = W, g 1 = V W V, is the symbol algebra of the canonical system on the first jet space for 2 dependent and s independent variables. Namely, there exists a coframe {ϖ 1, ϖ 2, ω 1,..., ω s, π 1 1,..., π s 1, π 1 2,..., π s 2} around x X such that D = {ϖ 1 = ϖ 2 = 0}, D = {ϖ 1 = ϖ 2 = ω 1 = = ω s = 0}, and { dϖ1 π 1 1 ω 1 + + π s 1 ω s (mod ϖ 1, ϖ 2 ) dϖ 2 π 1 2 ω 1 + + π s 2 ω s (mod ϖ 1, ϖ 2 ) Thus (X, D) satisfies the conditions (X.1) to (X.3) in 4.3 [14]. Hence we can construct the P D manifold (R(X); DX 1, D2 X ) as follows: Let us consider the collection R(X) of hyperplanes v in each tangent space T x (X) at x X which contains the fibre D(x) of the derived system D of D. R(X) = R x J(X, 3s + 1), x X R x = {v Gr(T x (X), 3s + 1) v D(x)} = P(T x (X)/ D(x)) = P 1. Moreover DX 1 is the canonical system obtained by the Grassmaniann construction and D2 X is the lift of D. In fact, DX 1 and D2 X are given by D 1 X(v) = ρ 1 (v) DX(v) 2 = ρ 1 (D(x)), for each v R(X) and x = ρ(v), where ρ : R(X) X is the projection (see 6.2[14] for the precise argument). Remark 2.1. Let c 1 (s, t) = g 2 g 1, g 2 = W, g 1 = V W V be the symbol algebra of the canonical system on the first order jet space for t dependent and s independent variables, where t = dim W and s = dim V. Let (Y, C) be a regular differential system of type c 1 (s, t). Let F (y) be the subspace of C(y) corresponding to W V under the symbol algebra identification at y Y. Then, when t 2, F is well defined subbundle of C (a covariant system) and (Y, C) is isomorphic with the canonical system of the first order jet space if and only if F is completely integrable (see Proposition 1.5 [10]). Moreover, when t 3, F is always completely integrable (Theorem 1.6 [10]). So c 1 (s, 2) case is very special and the Goursat gradations give special structures for F = D such that C = D. 3

Moreover, when (X, D) is the model space (M g, D g ) of type (X l, {α G }), R(X) can be identified with the model space (R g, E g ) of type (X l, {α θ, α G }) as follows (here, we understand α G denotes two simple roots α 1 and α 3 in case of BD l types and three simple roots α 1,α 3 and α 4 in case of D 4 ): Let (J g, C g ) be the standard contact manifold of type (X l, {α θ }). Then we have the double fibration; R g π g π c Jg M g Here (X l, {α θ, α G }) is a graded Lie algebra of depth 5 and satisfies the following: dim ǧ 5 = dim ǧ 4 = 1, dim ǧ 3 = dim ǧ 2 = s and dim ǧ 1 = s+1. In fact, comparing with the gradation of (X l, {α G }), we can check that ˇΦ + 5 = {θ}, ˇΦ + 4 = {θ α θ }, ˇΦ + 3 = Φ + 2, ˇΦ + 2 consists of roots θ β for each β ˇΦ + 3 and ˇΦ + 1 consists of roots α θ and θ α θ β for each β ˇΦ + 3 (see 6.2 [14] for BD l -type). Thus we see that (3) E g = (π c ) 1 (C g ), (2) E g = (π g ) 1 ( D g ) and E g = (π g ) 1 (D g ). We put D 1 = (3) E g and D 2 = E g. Then (R g ; D 1, D 2 ) is a P D manifold of second order. In fact, we have an isomorphism of (R g ; D 1, D 2 ) onto (R(M g ); DM 1 g, DM 2 g ) by the Realization Lemma for (R g, D 1, π g, M g ) and an embedding of R g into L(J g ) by the Realization Lemma for (R g, D 2, π c, J g ). Thus R g is identified with a R-space orbit in L(J g ). Moreover, putting C = (3) E g and N = (2) E g, (R g ; C, N) is an IG-manifold of corank 1, which is the global model of (W ; C, N) below. Let (X, D) be a regular differential system of type m, where m is the Goursat gradation of type X l. Then (X, D) is a regular differential system of type c 1 (s, 2). Hence, from (X, D), we can construct an IG manifold (W (X); C, N) of corank 1 and a P D manifold (R(X); D 1 X, D2 X ) of second order of Goursat type as is explained in 5 [13] or Theorem 6.1 [15]. Thus, from the standard differential system of type m, we can obtain the single equation of Grousat type (B l ) as in 5 (see Remark 6.2 (1) and 8.3 [15] for BD l -type). 2.3. Goursat gradations of Exceptional simple Lie algebras. Let m = g 3 g 2 g 1 be the Goursat gradation of type X l, where X l is one of the excetional simple Lie algebra. In order to obtain the structure of m, we will first check, in each case in 4, the following description in terms of the root space decomposition of m: Φ + 3 = {θ, θ α θ }, Φ + 2 = {β 1,..., β s }, Φ + 1 = {α θ + γ 1,..., α θ + γ s, γ 1,..., γ s }. where θ (2α θ + 3α G ), β i (α θ + 2α G ) and γ i α G are spanned by simple roots other than α θ and α G of X l such that θ = β i + α θ + γ i (i = 1,..., s). Then we will calculate the structure of m explicitly by use of the Chevalley basis of X l. By adjusting the Chevalley basis (especially, for E 6, E 7, E 8, by changing the orientation of Chevalley basis) suitably (see 4 for detail), we obtain the basis {W 1, W 2, Z 1,..., Z s, Y 1,..., Y s, X 1,..., X s } of m satisfying the following: such that g 3 = {W 1, W 2 }, g 2 = {Z 1,..., Z s }, g 1 = {Y 1,..., Y s, X 1,..., X s } [Z i, Y j ] = δ i jw 1, [Z i, X j ] = δ i jw 2 [X i, X j ] = [Y i, Y j ] = 0 (1 i, j s) In these basis, we calculate [X j, Y k ] for 1 j, k s in 4. 4

3. Classical Cases (BD l -type) The structure of the Goursat gradation m of type BD l, m = g 3 g 2 g 1 is given by the following brackets among the basis {W 1, W 2, Z 1,..., Z, Y 1,..., Y, X 1,..., X } of m ; such that g 3 = {W 1, W 2 }, g 2 = {Z 1,..., Z }, g 1 = {Y 1,..., Y, X 1,..., X } (3.1) [Z i, Y j ] = δ i jw 1, [Z i, X j ] = δ i jw 2, [X k1, Y k2 ] = δ k 1 k 2 Z 1, [X 1, Y k ] = [X k, Y 1 ] = Z k, [X i, X j ] = [Y i, Y j ] = 0 (1 i, j p + 1, 2 k, k 1, k 2 p + 1), In fact we obtain these relations through the matrices description of the Goursat gradation of o(p + 3, 3) (see 6.2 [14] and 8.3 [15]). The following differential system (X, D) on X = C 3p+5 describes the standard differential system of type m where D = { ϖ 1 = ϖ 2 = ω 1 = = ω = 0 }, ϖ 1 = dw 1 (z 1 + 1 x 2 k y k ) dy 1 {z k + 1(x 2 ky 1 + x 1 y k )} dy k, ϖ 2 = dw 2 (z 1 1 x 2 k y k ) dx 1 {z k 1(x 2 ky 1 + x 1 y k )} dx k, ω 1 = dz 1 + 1 (y 2 k dx k x k dy k ), ω k = dz k + 1(y 2 1 dx k x k dy 1 ) + 1(y 2 k dx 1 x 1 dy k ), (2 k p + 1). and (w 1, w 2, z 1,..., z, y 1,..., y, x 1,..., x ) is a coordinate system of X = C 3p+5. In fact we have dϖ 1 = dy 1 ω 1 + + dy ω, dϖ 2 = dx 1 ω 1 + + dx ω, (3.2) dω 1 = dy 2 dx 2 + + dy dx, dω k = dy 1 dx k + dy k dx 1 (2 k p + 1), which is the dual of (3.1). In particular, we have D = {ϖ 1 = ϖ 2 = 0} Now, utilizing the First Reduction Theorem, we will construct the model equation (A) from the standard differential system (X, D) of type m constructed as above, which is the local model corresponding to (BD l, {α 1, α 3 }). As in 2.2, (R(X); DX 1, D2 X ) is constructed as follows; R = R(X) is the collection of hyperplanes v in each tangent space T x (X) at x X which contains the fobre D of D. R(X) = x X R x J(X, 3p + 4), R x = {v Gr(T x (X), 3p + 4) v D(x)} = P 1, 5

Moreover D 1 is the canonical system obtained by the Grassmaniann construction and D 2 is the lift of D. Precisely, D 1 and D 2 are given by D 1 (v) = ρ 1 (v) D 2 (v) = ρ 1 (D(x)), for each v R(X) and x = ρ(v), where ρ : R(X) X is the projection. We introduce a fibre coordinate λ by ϖ = ϖ 1 + λϖ 2, where D 1 = { ϖ = 0 } and D = {ϖ 1 = ϖ 2 = 0}. Here (w 1, w 2, z 1,..., z, y 1,..., y, x 1,..., x, λ) constitute a coordinate system on R(X). Then we have dϖ = (dy 1 + λdx 1 ) ω 1 + + (dy + λdx ) ω + dλ ϖ 2, Ch (D 1 ) = { ϖ = ϖ 2 = ω 1 = = ω = dy 1 + λdx 1 = = dy + λdx = dλ = 0 }, D 2 = { ϖ = ϖ 2 = ω 1 = = ω = 0 } and D 2 = { ϖ = ϖ 2 = 0 }. Thus (R(X); D 1, D 2 ) is a P D-manifold of seconod rder. Now we calculate ϖ = ϖ 1 + λϖ 2 = dw 1 (z 1 + 1 x 2 k y k ) dy 1 {z k + 1(x 2 ky 1 + x 1 y k )} dy k, + λ{dw 2 (z 1 1 x 2 k y k ) dx 1 {z k 1(x 2 ky 1 + x 1 y k )} dx k }, = dw 1 + λdw 2 (z 1 + 1 x 2 k y k ) (dy 1 + λdx 1 ) {z k + 1(x 2 ky 1 + x 1 y k )} (dy k + λdx k ) + λ{( x k y k )dx 1 + (x k y 1 + x 1 y k )dx k } = d(w 1 + λw 2 ) (z 1 + 1 x 2 k y k ) d(y 1 + λx 1 ) {z k + 1(x 2 ky 1 + x 1 y k )} d(y k + λx k ) {w 2 (z 1 + 1 x 2 k y k ) x 1 {z k + 1(x 2 ky 1 + x 1 y k )} x k }dλ + λ{( x k y k )dx 1 + (x k y 1 + x 1 y k )dx k } Moreover we calculate λ{( x k y k )dx 1 + (x k y 1 + x 1 y k )dx k } = {λy k d(x 1 x k ) + 1λy 2 1d(x k ) 2 } = {d(λx 1 x k y k + 1 2 λx2 ky 1 ) x 1 x k d(λy k ) 1 2 x2 kd(λy 1 )} 6

Thus we obtain where Thus {d(λx 1 x k y k + 1 2 λx2 ky 1 ) (x 1 x k y k + 1 2 x2 ky 1 )dλ λx 1 x k dy k 1 2 λx2 kdy 1 } = = {d(λx 1 x k y k + 1 2 λx2 ky 1 ) (x 1 x k y k + 1 2 x2 ky 1 )dλ λx 1 x k d(y k + λx k ) 1 2 λx2 kd(y 1 + λx 1 ) + λx 1 x k d(λx k ) + 1 2 λx2 kd(λx 1 )} = {d(λx 1 x k y k + 1 2 λx2 ky 1 + 1 2 λ2 x 1 x 2 k) (x 1 x k y k + 1 2 x2 ky 1 1λx 2 1x 2 k)dλ λx 1 x k d(y k + λx k ) 1 2 λx2 kd(y 1 + λx 1 )} p+2 ϖ = dz P 1 dx 1 P k dx k Z = w 1 + λ w 2 + λ(x 1 x ky k + 1 2 x2 k y 1) + 1 2 λ2 x 1 x2 k, P 1 = w 2 x iz i 1λ x 2 1 x2 k, P 2 = z 1 + 1 2 x ky k + 1λ 2 x2 k, P k+1 = z k + 1(x 2 ky 1 + x 1 y k ) + λx 1 x k, (2 k p + 1), X 1 = λ, X i+1 = y i + λx i (1 i p + 1). p+2 D 1 = { dz P i dx i = 0 }, and (X 1,..., X p+2, Z, P 1,..., P p+2 ) constitutes a canonical coordinate system on J = R(X)/Ch (D 1 ). Putting x i = a i, we solve (3.3) λ = X 1, y i = X i+1 a i X 1 (1 i p + 1), z 1 = P 2 1 2 a kx k+1, z k = P k+1 1(a 2 kx 2 + a 1 X k+1 ) (2 k p + 1), w 2 = P 1 + a 1 (P 2 1 2 a kx k+1 ) + a k{p k+1 1(a 2 kx 2 + a 1 X k+1 )} + 1a 2 1 = P 1 + a kp k+1 + 1a 2 1 a2 k X 1 1 2 Then we calculate ω 1 = dp 2 + 1 a 2 2 k dx 1 a k dx k+1, p=2 a2 k X 1 a2 k X 2 a 1a k X k+1. ω k = dp k+1 + a 1 a k dx 1 a k dx 2 a 1 dx k+1 (k = 2,..., p + 1), ϖ 2 = dp 1 + a k dp k+1 + 1 a 2 1 a 2 k dx 1 1 a 2 2 k dx 2 a 1 a k dx k+1, = a k ω k + dp 1 a 1 a 2 k dx 1 + 1 a 2 2 k dx 2 + a 1 a k dx k+1. 7

This implies R(X) is given by the following 1 (p + 1)(p + 2) + 1 equations; 2 P 22 = 0, P ij = δ ij P 33 (3 i, j p + 2), P 11 = P 33 P2,k+1, 2 P 12 = 1 P 2 2 2,k+1, P 1,k+1 = P 33 P 2,k+1 (2 k p + 1), in terms of the canonical coordinate (X 1,..., X p+2, Z, P 1,..., P p+2, P 11,..., P p+2,p+2 ) of L(J). 4. Exceptional Cases Let m = g 3 g 2 g 1 be the Goursat gradation of type X l, where X l is one of the excetional simple Lie algebra. As in 2.3, we choose the basis {W 1, W 2, Z 1,..., Z s, Y 1,..., Y s, X 1,..., X s } of m satisfying the following: such that g 3 = {W 1, W 2 }, g 2 = {Z 1,..., Z s }, g 1 = {Y 1,..., Y s, X 1,..., X s } [Z i, Y j ] = δ i jw 1, [Z i, X j ] = δ i jw 2 [X i, X j ] = [Y i, Y j ] = 0 (1 i, j s) Utilizing the bilinear forms f i (x 1,..., x s, y 1,..., y s ) for i = 1,..., s, which describes the brackets [X j, Y k ] (see the following subsections), we can describe the standard differential system of type m as follows: Let (w 1, w 2, z 1,..., z s, y 1,..., y s, x 1,..., x s )be the linear coordinate of X = m given by the above basis of m. Then (X, D) on X = C 3s+2 describes the standard differential system of type m D = { ϖ 1 = ϖ 2 = ω 1 = = ω s = 0 }, where and In fact we have ϖ 1 = dw 1 (z i + 1f 2 i) dy i, ϖ 2 = dw 2 and we can calculate in each case ω i = dz i 1 2 {f i(x k, dy k ) f i (dx k, y k )} (z i 1f 2 i) dx i dω i = f i (dx k dy k ) (i = 1,..., s) (i = 1,..., s). dϖ 1 = dy 1 ω 1 + dy 2 ω 2 + + dy s ω s, dϖ 2 = dx 1 ω 1 + dx 2 ω 2 + + dx s ω s, which describes the structure of m. In particular, we have D = {ϖ 1 = ϖ 2 = 0} By the First Reduction Theorem, as in 2.2, our model overdetermined system (R(X); D 1 X, D2 X ) is constructed from (X, D) as follows: Moreover D 1 X and D2 X R(X) = x X R x J(X, 3s + 1), R x = {v Gr(T x (X), 3s + 1) v D(x)} = P 1, are given by D 1 X(v) = ρ 1 (v) DX(v) 2 = ρ 1 (D(x)), for v R(X), x = ρ(v), and ρ : R(X) X is the projection. 8

We introduce a fibre coordinate λ by ϖ = ϖ 1 + λϖ 2, where D 1 = { ϖ = 0 } and D = {ϖ 1 = ϖ 2 = 0}. Here (w 1, w 2, z 1,..., z s, y 1,..., y s, x 1,..., x s, λ) constitutes a coordinate system on R(X). Now we put x i f i = 2 y i g i where g i (x 1,..., x s ) = g i (x k ) = 1f 2 i(x k, x k ). Then we can check in each case f i dx i = y i dg i, where g is a cubic polynomial in x 1,..., x s. Now, for ϖ = ϖ 1 + λϖ 2, we calculate g i dx i = dg, 3g = dϖ = (dy 1 + λdx 1 ) ω 1 + + (dy s + λdx s ) ω s + dλ ϖ 2, ϖ = ϖ 1 + λϖ 2 = dw 1 (z i + 1f 2 i)dy i + λ{dw 2 = dw 1 + λdw 2 (z i 1f 2 i)dx i } (z i + 1f 2 i)(dy i + λdx i ) + λ = d(w 1 + λw 2 ) {w 2 f i dx i (x i z i + 1x 2 i f i )}dλ Moreover we calculate λ f i dx i = λ y i dg i = λ = d(λ = d(λ y i g i ) ( x i g i, (z i + 1f 2 i)d(y i + λx i ) + λ {d(y i g i ) g i dy i } y i g i ) ( 1 2 y i g i )dλ λ g i dy i x i f i )dλ λ g i dy i f i dx i λ g i dy i = λ g i d(y i + λx i ) λ( x i g i )dλ λ 2 = λ = λ Thus we obtain g i d(y i + λx i ) 3gλdλ λ 2 dg g i d(y i + λx i ) λgdλ d(λ 2 g) s+1 ϖ = dz P i dx i g i dx i 9

where Z = w 1 + λ w 2 + λ s y i g i + λ 2 g P 1 = w 2 s x i z i λ g, P i+1 = z i + 1f 2 i + λ g i (i = 1,..., s), X 1 = λ, X i+1 = y i + λx i (i = 1,..., s), Hence we have D 1 = { dz P 1 dx 1 P s+1 dx s+1 = 0 }, and (X 1,..., X s+1, Z, P 1,..., P s+1 ) constitutes a canonical coordinate system on J = R(X)/Ch (D 1 ). Putting x i = a i (i = 1,..., s), we solve λ = X 1, y i = X i+1 a i X 1 (i = 1,..., s), From f i (a k, X k+1 ) = f i (a k, y k + λa k ) = f i (a k, y k ) + 2λg i (a k ), we have z i = P i+1 1 2 f i λg i = P i+1 1 2 f i(a k, X k+1 ) (i = 1,..., s) From s x k( 1 2 f k + λg k ) = s (y k + λx k )g k, we have w 2 = P 1 + a k P k+1 + ĝx 1 ĝ k X k+1 where ĝ = g(a 1,..., a s ), and ĝ k = g k (a 1,..., a s ) Moreover we calculate (k = 1,..., s). ω i = dz i 1{f 2 i(x k, dy k ) f i (dx k, y k )} = d{p i+1 1f 2 i(a k, X k+1 )} 1{f 2 i(a k, d(x k+1 a k X 1 )) f i (da k, X k+1 a k X 1 )} = dp i+1 f i (a k, dx k+1 ) + 1f 2 i(a k, a k )dx 1 = dp i+1 f i (a k, dx k+1 ) + g i (a k, a k )dx 1 (i = 1,..., s) ϖ 2 = dw 2 (z i 1f 2 i)dx i = d(p 1 + a k P k+1 + ĝx 1 ĝ k X k+1 ) P k+1 da k + f i dx i + X 1 dĝ = dp 1 + a k dp k+1 + ĝdx 1 ĝ k dx k+1 = dp 1 + ( = a k ω k + 2 a k ω k + dp 1 2ĝdX 1 + ĝ k dx k+1 3ĝdX 1 ) + ĝdx 1 ĝ k dx k+1. ĝ k dx k+1 10

Thus we obtain (4.1) ω i = dp i+1 f i (a k, dx k+1 ) + ĝ i dx 1 (i = 1,..., s), ϖ 2 = a k ω k + dp 1 2ĝdX 1 + ĝ k dx k+1, = dp 1 + a k dp k+1 + ĝdx 1 ĝ k dx k+1. We will utilize the above formulae to describe our model system for each X l. In the following subsections, we wil follow Bourbaki [2] for the numbering of simple roots and descriptions of positive roots. Let us take a Chevalley basis {x α (α Φ); h i (1 i l)} of the Exceptional simple Lie algebra of type X l and put y β = x β for β Φ + ( cf. Chapter VII [5]). We will describe the structure of the Goursat gradation m in terms of {y β } β Φ +. Moreover we will regard simple Lie algebras E 6 and E 7 as regular subalgebras of E 8 and utilize the root space decomposition of E 8 to describe the structure of the Goursat gradation of E 6 and E 7. 4.1. Goursat gradation and model system of type F 4. Let m be the Goursat gradation of type F 4. For (F 4, {α 2 }), we have Φ + 3 = {β 24 = 2342, β 23 = 1342}, Φ + 2 = {β 22 = 1242, β 21 = 1232, β 20 = 1222, β 19 = 1231, β 18 = 1221, β 16 = 1220}, Φ + 1 = {β 5 = 1100, β 8 = 1110, β 11 = 1120, β 12 = 1111, β 15 = 1121, β 17 = 1122, β 2 = 0100, β 6 = 0110, β 9 = 0120, β 10 = 0111, β 13 = 0121, β 14 = 0122} where a 1 a 2 a 3 a 4 stands for the root β = 4 a iα i Φ +. We fix the orientation (or sign) of y β as in the following: First we choose the orientation of y αi for simple roots by fixing the root vectors y i = y αi g αi for i = 1, 2, 3, 4. For β i Φ + (i = 1,..., 24), we put y i = y βi and fix the orientation by the following order; y 5 = [y 1, y 2 ], y 6 = [y 2, y 3 ], y 8 = [y 3, y 5 ], 2y 9 = [y 3, y 6 ], y 10 = [y 4, y 6 ], y 11 = [y 1, y 9 ], y 12 = [y 4, y 8 ], y 13 = [y 3, y 10 ], 2y 14 = [y 4, y 13 ], y 15 = [y 4, y 11 ], y 16 = [y 2, y 11 ], y 17 = [y 2, y 14 ], y 18 = [y 4, y 16 ], y 19 = [y 3, y 18 ], y 20 = [y 2, y 17 ], y 21 = [y 4, y 19 ], 2y 22 = [y 3, y 21 ], y 23 = [y 2, y 22 ], y 24 = [y 1, y 23 ]. Then, for example, we calculate [y 1, y 6 ] = [y 1, [y 2, y 3 ]] = [[y 1, y 2 ], y 3 ] = [y 5, y 3 ] = y 8. In this way, by the repeated application of Jacobi identities, we obtain and Thus, putting 2y 24 = [ 2y 22, y 5 ] = [y 21, y 8 ] = [2y 20, y 11 ] = [y 19, y 12 ] = [ y 18, y 15 ] = [2y 16, y 17 ]. 2y 23 = [ 2y 22, y 2 ] = [y 21, y 6 ] = [2y 20, y 9 ] = [y 19, y 10 ] = [ y 18, y 13 ] = [2y 16, y 14 ]. W 1 = 2y 24, W 2 = 2y 23, Z 1 = 2y 22, Z 2 = y 21, Z 3 = 2y 20, Z 4 = y 19, Z 5 = y 18, Z 6 = 2y 16, Y 1 = y 5, Y 2 = y 8, Y 3 = y 11, Y 4 = y 12, Y 5 = y 15, Y 6 = y 17, X 1 = y 2, X 2 = y 6, X 3 = y 9, X 4 = y 10, X 5 = y 13, X 6 = y 14 11

we obtain the basis {W 1, W 2, Z 1,..., Z 6, Y 1,..., Y 6, X 1,..., X 6 } of m satisfying the following: such that g 3 = {W 1, W 2 }, g 2 = {Z 1,..., Z 6 }, g 1 = {Y 1,..., Y 6, X 1,..., X 6 } [Z i, Y j ] = δ i jw 1, [Z i, X j ] = δ i jw 2 [X i, X j ] = [Y i, Y j ] = 0 (1 i, j 6) Then we calculate [X j, Y k ] for 1 j, k 6 and obtain Z 1 = 2[X 3, Y 6 ] = [X 5, Y 5 ] = 2[X 6, Y 3 ], Z 2 = [X 2, Y 6 ] = [X 4, Y 5 ] = [X 5, Y 4 ] = [X 6, Y 2 ], Z 3 = 2[X 1, Y 6 ] = [X 4, Y 4 ] = 2[X 6, Y 1 ], Z 4 = [X 2, Y 5 ] = [X 3, Y 4 ] = [X 4, Y 3 ] = [X 5, Y 2 ], Z 5 = [X 1, Y 5 ] = [X 2, Y 4 ] = [X 4, Y 2 ] = [X 5, Y 1 ], Z 6 = 2[X 1, Y 3 ] = [X 2, Y 2 ] = 2[X 3, Y 1 ] Here we define the bilinear forms f i (x 1,..., x 6, y 1,..., y 6 )(i = 1,..., 6) as follows; Moreover we put f 1 = 2x 3 y 6 x 5 y 5 + 2x 6 y 3, f 2 = x 2 y 6 + x 4 y 5 + x 5 y 4 + x 6 y 2, f 3 = 2x 1 y 6 + x 4 y 4 + 2x 6 y 1, f 4 = x 2 y 5 + x 3 y 4 + x 4 y 3 + x 5 y 2, f 5 = x 1 y 5 + x 2 y 4 + x 4 y 2 x 5 y 1, f 6 = 2x 1 y 3 + x 2 y 2 + 2x 3 y 1, 6 x i f i = 2 6 y i g i where the quadratic forms g i (x 1,..., x 6 )(i = 1,..., 6) are given by Then we have { g 1 = 2x 3 x 6 1 2 x2 5, g 2 = x 2 x 6 + x 4 x 5, g 3 = 2x 1 x 6 + 1 2 x2 4, g 4 = x 2 x 5 + x 3 x 4, g 5 = x 1 x 5 + x 2 x 4, g 6 = 2x 1 x 3 + 1 2 x2 2, 6 f i dx i = 6 y i dg i, 6 g i dx i = dg, 3g = 6 x i g i, where g(x 1,..., x 6 ) is the cubic form given by g = 2x 1 x 3 x 6 + x 2 x 4 x 5 + 1 2 ( x 1 x 2 5 + x 2 2x 6 + x 3 x 2 4). 12

Thus, by (4.1), we obtain ω 1 = dp 2 + (2a 3 a 6 1 2 a2 5)dX 1 2a 6 dx 4 + a 5 dx 6 2a 3 dx 7, ω 2 = dp 3 + (a 2 a 6 + a 4 a 5 )dx 1 a 6 dx 3 a 5 dx 5 a 4 dx 6 a 2 dx 7, ω 3 = dp 4 + (2a 1 a 6 + 1 2 a2 4)dX 1 2a 6 dx 2 a 4 dx 5 2a 1 dx 7, ω 4 = dp 5 + (a 2 a 5 + a 3 a 4 )dx 1 a 5 dx 3 a 4 dx 4 a 3 dx 5 a 2 dx 6, ω 5 = dp 6 + ( a 1 a 5 + a 2 a 4 )dx 1 + a 5 dx 2 a 4 dx 3 a 2 dx 5 + a 1 dx 6, ω 6 = dp 7 + (2a 1 a 3 + 1 2 a2 2)dX 1 2a 3 dx 2 a 2 dx 3 2a 1 dx 4, ϖ 2 = a 1 ω 1 + a 2 ω 2 + + a 6 ω 6 + dp 1 (4a 1 a 3 a 6 + 2a 2 a 4 a 5 a 1 a 2 5 + a 2 2a 6 + a 3 a 2 4)dX 1 + (2a 3 a 6 1 2 a2 5)dX 2 + (a 2 a 6 + a 4 a 5 )dx 3 + (2a 1 a 6 + 1 2 a2 4)dX 4 + (a 2 a 5 + a 3 a 4 )dx 5 + ( a 1 a 5 + a 2 a 4 )dx 6 + (2a 1 a 3 + 1 2 a2 2)dX 7 This implies that our model system R(X) of type F 4 is given by the following 22 equations; P 66 = 1P 2 47 (= a 1 ), P 37 = P 56 (= a 2 ), P 55 = 1P 2 27 (= a 3 ), P 36 = P 45 (= a 4 ), P 35 = P 26 (= a 5 ), P 33 = 1P 2 24 (= a 6 ), P 22 = P 23 = P 25 = P 34 = P 44 = P 46 = P 57 = P 67 = P 77 = 0, P 17 = 2P 55 P 66 1P 2 2 37, P 16 = P 35 P 66 P 36 P 37, P 15 = P 35 P 37 P 36 P 55, P 14 = 2P 33 P 66 1P 2 2 36, P 13 = P 33 P 37 P 35 P 36, P 12 = 1P 2 2 35 2P 33 P 55, P 11 = 4P 33 P 55 P 66 + 2P 35 P 36 P 37 + P 2 35P 66 + P 33 P 2 37 + P 2 36P 55 in terms of the canonical coordinate (X 1,..., X 7, Z, P 1,..., P 7, P 11,..., P 77 ) of L(J). 4.2. Goursat gradation and model system of type E 6. In the following subsections, let us fix the root space decomposition of Simple Lie algebra E 8 and regard E 6 and E 7 as the regular subalgebras of E 8. Let m be the Goursat gradation of type E 6. For (E 6, {α 4 }), we have Φ + 3 = {γ 38 = 1 2 3 2 2 1, γ 37 = 1 2 3 1 2 1 }, Φ + 2 = {γ 36 = 1 2 2 1 2 1, γ 35 = 1 2 2 1 1 1, γ 34 = 1 1 2 1 2 1, γ 33 = 1 1 2 1 1 1, γ 32 = 1 2 2 1 1 0, γ 31 = 0 1 2 1 2 1, γ 29 = 1 1 2 1 1 0, γ 28 = 0 1 2 1 1 1, γ 24 = 0 1 2 1 1 0 }, Φ + 1 = {γ 10 = 0 0 1 1 0 0, γ 14 = 0 0 1 1 1 0, γ 15 = 0 1 1 1 0 0, γ 19 = 0 1 1 1 1 0, γ 20 = 0 0 1 1 1 1, γ 21 = 1 1 1 1 0 0, γ 25 = 0 1 1 1 1 1, γ 26 = 1 1 1 1 1 0, γ 30 = 1 1 1 1 1 1 γ 4 = 0 0 1 0 0 0, γ 11 = 0 0 1 0 1 0, γ 12 = 0 1 1 0 0 0, γ 16 = 0 1 1 0 1 0, γ 17 = 0 0 1 0 1 1, γ 18 = 1 1 1 0 0 0, γ 22 = 0 1 1 0 1 1, γ 23 = 1 1 1 0 1 0, γ 27 = 1 1 1 0 1 1 } where a 1 a 3 a 4 a 5 a 6 a 2 stands for the root γ = 6 a iα i Φ +. Also we put γ 9 = 0 0 0 0 1 1 and γ 13 = 1 1 0 0 0 0 for later use. We fix the orientation (or sign) of y γ as in the following: First we choose the orientation of y αi for simple roots by fixing the root vectors y i = y αi g αi for i = 1,..., 8. For γ i Φ + (i = 13

1,..., 38), we put y i = y γi and fix the orientation by the following order; y 9 = [y 5, y 6 ], y 10 = [y 2, y 4 ], y 11 = [y 4, y 5 ], y 12 = [y 3, y 4 ], y 13 = [y 1, y 3 ], y 14 = [y 2, y 11 ], y 15 = [y 2, y 12 ], y 16 = [y 3, y 11 ], y 17 = [y 4, y 9 ], y 18 = [y 4, y 13 ], y 19 = [y 2, y 16 ], y 20 = [y 2, y 17 ], y 21 = [y 2, y 18 ], y 22 = [y 3, y 17 ], y 23 = [y 1, y 16 ], y 24 = [y 4, y 19 ], y 25 = [y 2, y 22 ], y 26 = [y 2, y 23 ], y 27 = [y 1, y 22 ], y 28 = [y 4, y 25 ], y 29 = [y 2, y 26 ], y 30 = [y 2, y 27 ], y 31 = [y 5, y 28 ], y 32 = [y 3, y 29 ], y 33 = [y 4, y 30 ], y 34 = [y 1, y 31 ], y 35 = [y 3, y 33 ], y 36 = [y 3, y 34 ], y 37 = [y 4, y 36 ], y 38 = [y 2, y 37 ], Then, by the repeated application of Jacobi identities, we obtain y 38 = [ y 36, y 10 ] = [ y 35, y 14 ] = [y 34, y 15 ] = [y 33, y 19 ] = [y 32, y 20 ] = [y 31, y 21 ] = [ y 29, y 25 ] = [ y 28, y 26 ] = [y 24, y 30 ], y 37 = [ y 36, y 4 ] = [ y 35, y 11 ] = [y 34, y 12 ] = [y 33, y 16 ] = [y 32, y 17 ] = [y 31, y 18 ] = [ y 29, y 22 ] = [ y 28, y 23 ] = [y 24, y 27 ], Thus, putting W 1 = y 38, W 2 = y 37, Z 1 = y 36, Z 2 = y 35, Z 3 = y 34, Z 4 = y 33, Z 5 = y 32, Z 6 = y 31, Z 7 = y 29, Z 8 = y 28, Z 9 = y 24, Y 1 = y 10, Y 2 = y 14, Y 3 = y 15, Y 4 = y 19, Y 5 = y 20, Y 6 = y 21, Y 7 = y 25, Y 8 = y 26, Y 9 = y 30, X 1 = y 4, X 2 = y 11, X 3 = y 12, X 4 = y 16, X 5 = y 17, X 6 = y 18, X 7 = y 22, X 8 = y 23, X 9 = y 27, we obtain the basis {W 1, W 2, Z 1,..., Z 9, Y 1,..., Y 9, X 1,..., X 9 } of m satisfying the following: g 3 = {W 1, W 2 }, g 2 = {Z 1,..., Z 9 }, g 1 = {Y 1,..., Y 9, X 1,..., X 9 } such that [Z i, Y j ] = δjw i 1, [Z i, X j ] = δjw i 2 [X i, X j ] = [Y i, Y j ] = 0 (1 i, j 9) Then we calculate [X j, Y k ] for 1 j, k 9 and obtain Z 1 = [X 4, Y 9 ] = [X 7, Y 8 ] = [X 8, Y 7 ] = [X 9, Y 4 ], Z 2 = [X 3, Y 9 ] = [X 6, Y 7 ] = [X 7, Y 6 ] = [X 9, Y 3 ], Z 3 = [X 2, Y 9 ] = [X 5, Y 8 ] = [X 8, Y 5 ] = [X 9, Y 2 ], Z 4 = [X 1, Y 9 ] = [X 5, Y 6 ] = [X 6, Y 5 ] = [X 9, Y 1 ], Z 5 = [X 3, Y 8 ] = [X 4, Y 6 ] = [X 6, Y 4 ] = [X 8, Y 3 ], Z 6 = [X 2, Y 7 ] = [X 4, Y 5 ] = [X 5, Y 4 ] = [X 7, Y 2 ], Z 7 = [X 1, Y 8 ] = [X 2, Y 6 ] = [X 6, Y 2 ] = [X 8, Y 1 ], Z 8 = [X 1, Y 7 ] = [X 3, Y 5 ] = [X 5, Y 3 ] = [X 7, Y 1 ], Z 9 = [X 1, Y 4 ] = [X 2, Y 3 ] = [X 3, Y 2 ] = [X 4, Y 1 ]. 14

Here we define the bilinear forms f i (x 1,..., x 9, y 1,..., y 9 )(i = 1,..., 9) as follows; f 1 = x 4 y 9 x 7 y 8 x 8 y 7 + x 9 y 4, f 2 = x 3 y 9 x 6 y 7 x 7 y 6 x 9 y 3, f 3 = x 2 y 9 + x 5 y 8 + x 8 y 5 x 9 y 2, f 4 = x 1 y 9 + x 5 y 6 + x 6 y 5 + x 9 y 1, f 5 = x 3 y 8 + x 4 y 6 + x 6 y 4 + x 8 y 3, f 6 = x 2 y 7 + x 4 y 5 + x 5 y 4 x 7 y 2, f 7 = x 1 y 8 x 2 y 6 x 6 y 2 x 8 y 1, f 8 = x 1 y 7 + x 3 y 5 + x 5 y 3 x 7 y 1, f 9 = x 1 y 4 x 2 y 3 x 3 y 2 + x 4 y 1 Moreover we put 9 x i f i = 2 9 y i g i where the quadratic forms g i (x 1,..., x 9 )(i = 1,..., 9) are given by g 1 = x 4 x 9 x 7 x 8, g 2 = x 3 x 9 x 6 x 7, g 3 = x 5 x 8 x 2 x 9, g 4 = x 1 x 9 + x 5 x 6, g 5 = x 3 x 8 + x 4 x 6, g 6 = x 4 x 5 x 2 x 7, g 7 = x 1 x 8 x 2 x 6, g 8 = x 3 x 5 x 1 x 7, g 9 = x 1 x 4 x 2 x 3. Then we have 9 f i dx i = 9 y i dg i, where g(x 1,..., x 9 ) is the cubic form given by Thus, by (4.1), we obtain 9 g i dx i = dg, 3g = 9 x i g i, g = x 1 x 4 x 9 x 1 x 7 x 8 x 2 x 3 x 9 x 2 x 6 x 7 + x 3 x 5 x 8 + x 4 x 5 x 6. ω 1 = dp 2 + (a 4 a 9 a 7 a 8 )dx 1 a 9 dx 5 + a 8 dx 8 + a 7 dx 9 a 4 dx 10, ω 2 = dp 3 (a 3 a 9 + a 6 a 7 )dx 1 + a 9 dx 4 + a 7 dx 7 + a 6 dx 8 + a 3 dx 10, ω 3 = dp 4 + (a 5 a 8 a 2 a 9 )dx 1 + a 9 dx 3 a 8 dx 6 a 5 dx 9 + a 2 dx 10, ω 4 = dp 5 + (a 1 a 9 + a 5 a 6 )dx 1 a 9 dx 2 a 6 dx 6 a 5 dx 7 a 1 X 10, ω 5 = dp 6 + (a 3 a 8 + a 4 a 6 )dx 1 a 8 dx 4 a 6 dx 5 a 4 dx 7 a 3 dx 9, ω 6 = dp 7 + (a 4 a 5 a 2 a 7 )dx 1 + a 7 dx 3 a 5 dx 5 a 4 dx 6 + a 2 X 8, ω 7 = dp 8 (a 1 a 8 + a 2 a 6 )dx 1 + a 8 dx 2 + a 6 dx 3 + a 2 dx 7 + a 1 dx 9, ω 8 = dp 9 + (a 3 a 5 a 1 a 7 )dx 1 + a 7 dx 2 a 5 dx 4 a 3 dx 6 + a 1 dx 8, ω 9 = dp 10 + (a 1 a 4 a 2 a 3 )dx 1 a 4 dx 2 + a 3 dx 3 + a 2 dx 4 a 1 dx 5, ϖ 2 = a 1 ω 1 + a 2 ω 2 + + a 9 ω 9 + dp 1 2(a 1 a 4 a 9 a 1 a 7 a 8 a 2 a 3 a 9 a 2 a 6 a 7 + a 3 a 5 a 8 + a 4 a 5 a 6 )dx 1 + (a 4 a 9 a 7 a 8 )dx 2 (a 3 a 9 + a 6 a 7 )dx 3 + (a 5 a 8 a 2 a 9 )dx 4 + (a 1 a 9 + a 5 a 6 )dx 5 + (a 3 a 8 + a 4 a 6 )dx 6 + (a 4 a 5 a 2 a 7 )dx 7 (a 1 a 8 + a 2 a 6 )dx 8 + (a 3 a 5 a 1 a 7 )dx 9 + (a 1 a 4 a 2 a 3 )dx 10 15

This implies R(X) is given by the following 46 equations; P 5,10 = P 89 (= a 1 ), P 78 = P 4,10 (= a 2 ), P 69 = P 3,10 (= a 3 ), P 67 = P 2,10 (= a 4 ), P 49 = P 57 (= a 5 ), P 56 = P 38 (= a 6 ), P 29 = P 37 (= a 7 ), P 46 = P 28 (= a 8 ), P 25 = P 34 (= a 9 ) P 22 = P 23 = P 24 = P 26 = P 27 = P 33 = P 35 = P 36 = P 39 = P 44 = P 45 = P 47 = P 48 = P 55 = 0, P 58 = P 59 = P 66 = P 68 = P 6,10 = P 77 = P 79 = P 7,10 = P 88 = P 8,10 = P 99 = P 9,10 = P 10,10 = 0, P 1,10 = P 78 P 69 P 5,10 P 67, P 19 = P 5,10 P 29 P 69 P 49, P 18 = P 5,10 P 46 P 78 P 56, P 17 = P 78 P 29 P 67 P 49, P 16 = P 69 P 46 P 67 P 56, P 15 = P 5,10 P 25 P 49 P 56, P 14 = P 78 P 25 P 49 P 46, P 13 = P 69 P 25 P 56 P 29, P 12 = P 29 P 46 P 67 P 25, P 11 = 2(P 5,10 P 67 P 25 + P 5,10 P 29 P 46 + P 78 P 69 P 25 P 78 P 56 P 29 + P 69 P 49 P 46 + P 67 P 49 P 56 ). in coordinates (X 1,..., X 10, Z, P 1,..., P 10, P 11,..., P 10,10 ) of L(J). 4.3. Goursat gradation and model system of type E 7. Let m be the Goursat gradation of type E 7. For (E 7, {α 3 }), we have Φ + 3 = {γ 64 = 2 3 4 2 3 2 1, γ 63 = 1 3 4 2 3 2 1 }, Φ 2 + = {γ 62 = 1 2 4 2 3 2 1, γ 61 = 1 2 3 2 3 2 1, γ 60 = 1 2 3 1 3 2 1, γ 59 = 1 2 3 2 2 2 1, γ 58 = 1 2 3 1 2 2 1, γ 57 = 1 2 3 2 2 1 1, γ 56 = 1 2 2 1 2 2 1, γ 55 = 1 2 3 1 2 1 1, γ 53 = 1 2 2 1 2 1 1, γ 50 = 1 2 2 1 1 1 1, γ 38 = 1 2 3 2 2 1 0, γ 37 = 1 2 3 1 2 1 0, γ 36 = 1 2 2 1 2 1 0, γ 35 = 1 2 2 1 1 1 0, γ 32 = 1 2 2 1 1 0 0, }, Φ + 1 = {γ 13 = 1 1 0 0 0 0 0, γ 18 = 1 1 1 0 0 0 0, γ 21 = 1 1 1 1 0 0 0, γ 23 = 1 1 1 0 1 0 0, γ 26 = 1 1 1 1 1 0 0, γ 27 = 1 1 1 0 1 1 0, γ 29 = 1 1 2 1 1 0 0, γ 30 = 1 1 1 1 1 1 0, γ 33 = 1 1 2 1 1 1 0, γ 34 = 1 1 2 1 2 1 0, γ 45 = 1 1 1 0 1 1 1, γ 47 = 1 1 1 1 1 1 1, γ 48 = 1 1 2 1 1 1 1, γ 51 = 1 1 2 1 2 1 1, γ 54 = 1 1 2 1 2 2 1, γ 3 = 0 1 0 0 0 0 0, γ 12 = 0 1 1 0 0 0 0, γ 15 = 0 1 1 1 0 0 0, γ 16 = 0 1 1 0 1 0 0, γ 19 = 0 1 1 1 1 0 0, γ 22 = 0 1 1 0 1 1 0, γ 24 = 0 1 2 1 1 0 0, γ 25 = 0 1 1 1 1 1 0, γ 28 = 0 1 2 1 1 1 0, γ 31 = 0 1 2 1 2 1 0, γ 43 = 0 1 1 0 1 1 1, γ 44 = 0 1 1 1 1 1 1, γ 46 = 0 1 2 1 1 1 1, γ 49 = 0 1 2 1 2 1 1, γ 52 = 0 1 2 1 2 2 1, }, where a 1 a 3 a 4 a 5 a 6 a 7 a 2 stands for the root γ = 7 a iα i Φ +. We fix the orientation (or sign) of y γ as in the following: For γ i Φ + (i = 39,..., 64), we put y i = y γi and fix the orientation by the following order; y 39 = [y 6, y 11 ], y 40 = [y 5, y 39 ], y 41 = [y 4, y 40 ], y 42 = [y 2, y 41 ], y 43 = [y 7, y 22 ], y 44 = [y 2, y 43 ], y 45 = [y 7, y 27 ], y 46 = [y 4, y 44 ], y 47 = [y 2, y 45 ], y 48 = [y 4, y 47 ], y 49 = [y 7, y 31 ], y 50 = [y 7, y 35 ], y 51 = [y 7, y 34 ], y 52 = [y 6, y 49 ], y 53 = [y 5, y 50 ], y 54 = [y 6, y 51 ], y 55 = [y 4, y 53 ], y 56 = [y 6, y 53 ], y 57 = [y 2, y 55 ], y 58 = [y 4, y 56 ], y 59 = [y 2, y 58 ], y 60 = [y 5, y 58 ], y 61 = [y 2, y 60 ], y 62 = [y 4, y 61 ], y 63 = [y 3, y 62 ], y 64 = [y 1, y 63 ], Then, by the repeated application of Jacobi identities, we obtain y 64 = [ y 62, y 13 ] = [y 61, y 18 ] = [y 60, y 21 ] = [ y 59, y 23 ] = [y 58, y 26 ] = [ y 57, y 27 ] = [ y 56, y 29 ] = [y 55, y 30 ] = [ y 53, y 33 ] = [y 50, y 34 ] = [y 38, y 45 ] = [ y 37, y 47 ] = [y 36, y 48 ] = [ y 35, y 51 ] = [ y 32, y 54 ] y 63 = [ y 62, y 3 ] = [y 61, y 12 ] = [y 60, y 15 ] = [ y 59, y 16 ] = [y 58, y 19 ] = [ y 57, y 22 ] = [ y 56, y 24 ] = [y 55, y 25 ] = [ y 53, y 28 ] = [y 50, y 31 ] = [y 38, y 43 ] = [ y 37, y 44 ] = [y 36, y 46 ] = [ y 35, y 49 ] = [ y 32, y 52 ] 16

Thus, putting W 1 = y 64, W 2 = y 63, Z 1 = y 62, Z 2 = y 61, Z 3 = y 60, Z 4 = y 59, Z 5 = y 58, Z 6 = y 57, Z 7 = y 56, Z 8 = y 55, Z 9 = y 53, Z 10 = y 50, Z 11 = y 38, Z 12 = y 37, Z 13 = y 36, Z 14 = y 35, Z 15 = y 32, Y 1 = y 13, Y 2 = y 18, Y 3 = y 21, Y 4 = y 23, Y 5 = y 26, Y 6 = y 27, Y 7 = y 29, Y 8 = y 30, Y 9 = y 33, Y 10 = y 34, Y 11 = y 45, Y 12 = y 47, Y 13 = y 48, Y 14 = y 51, Y 15 = y 54, X 1 = y 3, X 2 = y 12, X 3 = y 15, X 4 = y 16, X 5 = y 19, X 6 = y 22, X 7 = y 24, X 8 = y 25, X 9 = y 28, X 10 = y 31, X 11 = y 43, X 12 = y 44, X 13 = y 46, X 14 = y 49, X 15 = y 52. we obtain the basis {W 1, W 2, Z 1,..., Z 15, Y 1,..., Y 15, X 1,..., X 15 } of m satisfying the following: g 3 = {W 1, W 2 }, g 2 = {Z 1,..., Z 15 }, g 1 = {Y 1,..., Y 15, X 1,..., X 15 } such that [Z i, Y j ] = δ i jw 1, [Z i, X j ] = δ i jw 2 [X i, X j ] = [Y i, Y j ] = 0 (1 i, j 15) Then we calculate [X j, Y k ] for 1 j, k 15 and obtain Z 1 = [X 7, Y 15 ] = [X 9, Y 14 ] = [X 10, Y 13 ] = [X 13, Y 10 ] = [X 14, Y 9 ] = [X 15, Y 7 ] Z 2 = [X 5, Y 15 ] = [X 8, Y 14 ] = [X 10, Y 12 ] = [X 12, Y 10 ] = [X 14, Y 8 ] = [X 15, Y 5 ], Z 3 = [X 4, Y 15 ] = [X 6, Y 14 ] = [X 10, Y 11 ] = [X 11, Y 10 ] = [X 14, Y 6 ] = [X 15, Y 4 ], Z 4 = [X 3, Y 15 ] = [X 8, Y 13 ] = [X 9, Y 12 ] = [X 12, Y 9 ] = [X 13, Y 8 ] = [X 15, Y 3 ], Z 5 = [X 2, Y 15 ] = [X 6, Y 13 ] = [X 9, Y 11 ] = [X 11, Y 9 ] = [X 13, Y 6 ] = [X 15, Y 2 ], Z 6 = [X 3, Y 14 ] = [X 5, Y 13 ] = [X 7, Y 12 ] = [X 12, Y 7 ] = [X 13, Y 5 ] = [X 14, Y 3 ], Z 7 = [X 1, Y 15 ] = [X 6, Y 12 ] = [X 8, Y 11 ] = [X 11, Y 8 ] = [X 12, Y 6 ] = [X 15, Y 1 ] Z 8 = [X 2, Y 14 ] = [X 4, Y 13 ] = [X 7, Y 11 ] = [X 11, Y 7 ] = [X 13, Y 4 ] = [X 14, Y 2 ], Z 9 = [X 1, Y 14 ] = [X 4, Y 12 ] = [X 5, Y 11 ] = [X 11, Y 5 ] = [X 12, Y 4 ] = [X 14 Y 1 ] Z 10 = [X 1, Y 13 ] = [X 2, Y 12 ] = [X 3, Y 11 ] = [X 11, Y 3 ] = [X 12, Y 2 ] = [X 13, Y 1 ] Z 11 = [X 3, Y 10 ] = [X 5, Y 9 ] = [X 7, Y 8 ] = [X 8, Y 7 ] = [X 9, Y 5 ] = [X 10, Y 3 ], Z 12 = [X 2, Y 10 ] = [X 4, Y 9 ] = [X 6, Y 7 ] = [X 7, Y 6 ] = [X 9, Y 4 ] = [X 10, Y 2 ] Z 13 = [X 1, Y 10 ] = [X 4, Y 8 ] = [X 5, Y 6 ] = [X 6, Y 5 ] = [X 8, Y 4 ] = [X 10, Y 1 ] Z 14 = [X 1, Y 9 ] = [X 2, Y 8 ] = [X 3, Y 6 ] = [X 6, Y 3 ] = [X 8, Y 2 ] = [X 9, Y 1 ] Z 15 = [X 1, Y 7 ] = [X 2, Y 5 ] = [X 3, Y 4 ] = [X 4, Y 3 ] = [X 5, Y 2 ] = [X 7, Y 1 ], 17

Here we define the bilinear forms f i (x 1,..., x 15, y 1,..., y 15 )(i = 1,..., 15) as follows; f 1 = x 7 y 15 x 9 y 14 + x 10 y 13 + x 13 y 10 x 14 y 9 x 15 y 7, f 2 = x 5 y 15 + x 8 y 14 x 10 y 12 x 12 y 10 + x 14 y 8 + x 15 y 5, f 3 = x 4 y 15 + x 6 y 14 x 10 y 11 x 11 y 10 + x 14 y 6 + x 15 y 4, f 4 = x 3 y 15 x 8 y 13 + x 9 y 12 + x 12 y 9 x 13 y 8 + x 15 y 3, f 5 = x 2 y 15 + x 6 y 13 x 9 y 11 x 11 y 9 + x 13 y 6 + x 15 y 2, f 6 = x 3 y 14 + x 5 y 13 x 7 y 12 x 12 y 7 + x 13 y 5 + x 14 y 3, f 7 = x 1 y 15 x 6 y 12 + x 8 y 11 + x 11 y 8 x 12 y 6 x 15 y 1 f 8 = x 2 y 14 x 4 y 13 + x 7 y 11 + x 11 y 7 x 13 y 4 + x 14 y 2, f 9 = x 1 y 14 + x 4 y 12 x 5 y 11 x 11 y 5 + x 12 y 4 x 14 y 1 f 10 = x 1 y 13 x 2 y 12 x 3 y 11 x 11 y 3 x 12 y 2 + x 13 y 1 f 11 = x 3 y 10 x 5 y 9 + x 7 y 8 + x 8 y 7 x 9 y 5 x 10 y 3, f 12 = x 2 y 10 + x 4 y 9 x 6 y 7 x 7 y 6 + x 9 y 4 x 10 y 2 f 13 = x 1 y 10 x 4 y 8 + x 5 y 6 + x 6 y 5 x 8 y 4 + x 10 y 1 f 14 = x 1 y 9 + x 2 y 8 + x 3 y 6 + x 6 y 3 + x 8 y 2 x 9 y 1 f 15 = x 1 y 7 + x 2 y 5 + x 3 y 4 + x 4 y 3 + x 5 y 2 x 7 y 1 Moreover we put 15 15 x i f i = 2 y i g i where the quadratic forms g i (x 1,..., x 15 )(i = 1,..., 15) are given by g 1 = x 7 x 15 x 9 x 14 + x 10 x 13, g 2 = x 5 x 15 + x 8 x 14 x 10 x 12, g 3 = x 4 x 15 + x 6 x 14 x 10 x 11, g 4 = x 3 x 15 x 8 x 13 + x 9 x 12, g 5 = x 2 x 15 + x 6 x 13 x 9 x 11, g 6 = x 3 x 14 + x 5 x 13 x 7 x 12, g 7 = x 1 x 15 x 6 x 12 + x 8 x 11, g 8 = x 2 x 14 x 4 x 13 + x 7 x 11, g 9 = x 1 x 14 + x 4 x 12 x 5 x 11, g 10 = x 1 x 13 x 2 x 12 x 3 x 11, g 11 = x 3 x 10 x 5 x 9 + x 7 x 8, g 12 = x 2 x 10 + x 4 x 9 x 6 x 7, g 13 = x 1 x 10 x 4 x 8 + x 5 x 6, g 14 = x 1 x 9 + x 2 x 8 + x 3 x 6, g 15 = x 1 x 7 + x 2 x 5 + x 3 x 4. Then we have 15 15 f i dx i = y i dg i, 15 15 g i dx i = dg, 3g = x i g i, where g(x 1,..., x 15 ) is the cubic form given by g = x 1 x 7 x 15 x 1 x 9 x 14 + x 1 x 10 x 13 + x 2 x 5 x 15 + x 2 x 8 x 14 x 2 x 10 x 12. + x 3 x 4 x 15 + x 3 x 6 x 14 x 3 x 10 x 11 x 4 x 8 x 13 + x 4 x 9 x 12 + x 5 x 6 x 13 x 5 x 9 x 11 x 6 x 7 x 12 + x 7 x 8 x 11. 18

Thus, by (4.1), we obtain ω 1 = dp 2 + ĝ 1 dx 1 ( a 15 dx 8 a 14 dx 10 + a 13 dx 11 + a 10 dx 14 a 9 dx 15 a 7 dx 16 ), ω 2 = dp 3 + ĝ 2 dx 1 (a 15 dx 6 + a 14 dx 9 a 12 dx 11 a 10 dx 13 + a 8 dx 15 + a 5 dx 16 ), ω 3 = dp 4 + ĝ 3 dx 1 (a 15 dx 5 + a 14 dx 7 a 11 dx 11 a 10 dx 12 + a 6 dx 15 + a 4 dx 16 ), ω 4 = dp 5 + ĝ 4 dx 1 (a 15 dx 4 a 13 dx 9 + a 12 dx 10 + a 9 dx 13 a 8 dx 14 + a 3 dx 16 ), ω 5 = dp 6 + ĝ 5 dx 1 (a 15 dx 3 + a 13 dx 7 a 11 dx 10 a 9 X 12 + a 6 dx 14 + a 2 dx 16 ), ω 6 = dp 7 + ĝ 6 dx 1 (a 14 dx 4 + a 13 dx 6 a 12 dx 8 a 7 dx 13 + a 5 dx 14 + a 3 dx 15 ), ω 7 = dp 8 + ĝ 7 dx 1 ( a 15 dx 2 a 12 dx 7 + a 11 dx 9 + a 8 dx 12 a 6 dx 13 a 1 dx 16 ), ω 8 = dp 9 + ĝ 8 dx 1 (a 14 dx 3 a 13 dx 5 + a 11 dx 8 + a 7 dx 12 a 4 dx 14 + a 2 dx 15 ), ω 9 = dp 10 + ĝ 9 dx 1 ( a 14 dx 2 + a 12 dx 5 a 11 dx 6 a 5 dx 12 + a 4 dx 13 a 1 dx 15 ), ω 10 = dp 11 + ĝ 10 dx 1 (a 13 dx 2 a 12 dx 3 a 11 dx 4 a 3 dx 12 a 2 dx 13 + a 1 dx 14 ), ω 11 = dp 12 + ĝ 11 dx 1 ( a 10 dx 4 a 9 dx 6 + a 8 dx 8 + a 7 dx 9 a 5 dx 10 a 3 dx 11 ), ω 12 = dp 13 + ĝ 12 dx 1 ( a 10 dx 3 + a 9 dx 5 a 7 dx 7 a 6 dx 8 + a 4 dx 10 a 2 dx 11 ), ω 13 = dp 14 + ĝ 13 dx 1 (a 10 dx 2 a 8 dx 5 + a 6 dx 6 + a 5 dx 7 a 4 dx 9 + a 1 dx 11 ), ω 14 = dp 15 + ĝ 14 dx 1 ( a 9 dx 2 + a 8 dx 3 + a 6 dx 4 + a 3 dx 7 + a 2 dx 9 a 1 dx 10 ), ω 15 = dp 16 + ĝ 15 dx 1 ( a 7 dx 2 + a 5 dx 3 + a 4 dx 4 + a 3 dx 5 + a 2 dx 6 a 1 dx 8 ), ϖ 2 = a 1 ω 1 + a 2 ω 2 + + a 15 ω 15 + dp 1 2ĝdX 1 + ĝ 1 dx 2. + + ĝ 15 dx 16 This implies R(X) is given by the following 121 equations; P 8,16 = P 10,15 = P 11,14 (= a 1 ), P 6,16 = P 9,15 = P 11,13 (= a 2 ), P 5,16 = P 7,15 = P 11,12 (= a 3 ), P 4,16 = P 9,14 = P 10,13 (= a 4 ), P 3,16 = P 7,14 = P 10,12 (= a 5 ), P 4,15 = P 6,14 = P 8,13 (= a 6 ), P 2,16 = P 7,13 = P 9,12 (= a 7 ), P 3,15 = P 5,14 = P 8,12 (= a 8 ), P 2,15 = P 5,13 = P 6,12 (= a 9 ), P 2,14 = P 3,13 = P 4,12 (= a 10 ), P 4,11 = P 6,10 = P 8,9 (= a 11 ), P 3,11 = P 5,10 = P 7,8 (= a 12 ), P 2,11 = P 5,9 = P 6,7 (= a 13 ), P 2,10 = P 3,9 = P 4,7 (= a 14 ), P 2,8 = P 3,6 = P 4,5 (= a 15 ), P 2,2 = P 2,3 = P 2,4 = P 2,5 = P 2,6 = P 2,7 = P 2,9 = P 2,12 = P 2,13 = P 3,3 = P 3,4 = P 3,5 = P 3,7 = 0, P 3,8 = P 3,10 = P 3,12 = P 3,14 = P 4,4 = P 4,6 = P 4,8 = P 4,9 = P 4,10 = P 4,13 = P 4,14 = P 5,5 = 0, P 5,6 = P 5,7 = P 5,8 = P 5,11 = P 5,12 = P 5,15 = P 6,6 = P 6,8 = P 6,9 = P 6,11 = P 6,13 = P 6,15 = 0, P 7,7 = P 7,9 = P 7,10 = P 7,11 = P 7,12 = P 7,16 = P 8,8 = P 8,10 = P 8,11 = P 8,14 = P 8,15 = 0, P 9,9 = P 9,10 = P 9,11 = P 9,13 = P 9,16 = P 10,10 = P 10,11 = P 10,14 = P 10,16 = 0, P 11,11 = P 11,15 = P 11,16 = P 12,12 = P 12,13 = P 12,14 = P 12,15 = P 12,16 = 0, P 13,13 = P 13,14 = P 13,15 = P 13,16 = P 14,14 = P 14,15 = P 14,16 = P 15,15 = P 15,16 = P 16,16 = 0 P 1,16 = P 2,16 P 8,16 P 3,16 P 6,16 P 4,16 P 5,16, P 1,15 = P 2,15 P 8,16 P 3,15 P 6,16 P 4,15 P 5,16, P 1,14 = P 2,14 P 8,16 + P 3,15 P 4,16 P 3,16 P 4,15, P 1,13 = P 2,14 P 5,16 + P 2,15 P 4,16 P 2,16 P 4,15, P 1,12 = P 2,14 P 5,16 P 2,15 P 3,16 + P 2,15 P 2,16, P 1,11 = P 2,11 P 8,16 P 3,11 P 6,16 + P 4,11 P 5,16, P 1,10 = P 2,10 P 8,16 + P 3,11 P 4,16 P 3,16 P 4,11, P 1,9 = P 2,10 P 6,16 + P 2,11 P 4,16 P 2,16 P 4,11, P 1,8 = P 2,8 P 8,16 P 2,11 P 4,15 + P 3,15 P 8,9, P 1,7 = P 2,10 P 5,16 P 2,11 P 3,16 + P 2,16 P 3,11, P 1,6 = P 2,8 P 6,16 P 2,11 P 4,15 + P 2,16 P 4,11, P 1,5 = P 2,8 P 5,16 + P 2,11 P 3,15 P 2,15 P 3,11, P 1,4 = P 2,8 P 4,16 + P 2,10 P 4,15 P 2,14 P 4,11, P 1,3 = P 2,8 P 3,16 + P 2,10 P 3,15 P 2,14 P 3,11, P 1,2 = P 2,8 P 2,16 + P 2,10 P 2,15 P 2,11 P 2,14 19

P 1,1 = 2(P 2,8 P 2,16 P 8,16 + P 2,10 P 2,15 P 8,16 P 2,11 P 2,14 P 8,16 P 2,8 P 3,16 P 6,16 P 2,10 P 3,15 P 6,16 + P 2,14 P 3,11 P 6,16 P 2,8 P 4,16 P 5,16 P 2,10 P 4,15 P 5,16 + P 2,14 P 4,11 P 5,16 P 2,11 P 3,15 P 4,16 + P 2,15 P 3,11 P 4,16 + P 2,11 P 3,16 P 4,15 P 2,15 P 3,16 P 4,11 P 2,16 P 3,11 P 4,15 + P 2,16 P 3,15 P 4,11 ), in coordinates (X 1,..., X 16, Z, P 1,..., P 16, P 11,..., P 16,16 ) of L(J). 4.4. Goursat gradation and model system of type E 8. Let m be the Goursat gradation of type E 8. For (E 8, {α 7 }), we have Φ + 3 = {γ 120 = 2 4 6 3 5 4 3 2, γ 119 = 2 4 6 3 5 4 3 1 }, Φ + 2 = {γ 118 = 2 4 6 3 5 4 2 1, γ 117 = 2 4 6 3 5 3 2 1, γ 116 = 2 4 6 3 4 3 2 1, γ 115 = 2 4 5 3 4 3 2 1, γ 114 = 2 4 5 2 4 3 2 1, γ 113 = 2 3 5 3 4 3 2 1, γ 112 = 2 3 5 2 4 3 2 1, γ 111 = 1 3 5 3 4 3 2 1, γ 110 = 2 3 4 2 4 3 2 1, γ 109 = 1 3 5 2 4 3 2 1, γ 108 = 1 3 4 2 4 3 2 1, γ 107 = 2 3 4 2 3 3 2 1, γ 106 = 1 2 4 2 4 3 2 1, γ 105 = 1 3 4 2 3 3 2 1, γ 104 = 2 3 4 2 3 2 2 1, γ 103 = 1 2 4 2 3 3 2 1, γ 102 = 1 3 4 2 3 2 2 1, γ 100 = 1 2 3 2 3 3 2 1, γ 99 = 1 2 4 2 3 2 2 1, γ 97 = 1 2 3 1 3 3 2 1, γ 96 = 1 2 3 2 3 2 2 1, γ 94 = 1 2 3 1 3 2 2 1, γ 93 = 1 2 3 2 2 2 2 1, γ 91 = 1 2 3 1 2 2 2 1, γ 88 = 1 2 2 1 2 2 2 1, γ 85 = 1 1 2 1 2 2 2 1, γ 82 = 0 1 2 1 2 2 2 1, }, Φ + 1 = {γ 65 = 0 0 0 0 0 0 1 1, γ 66 = 0 0 0 0 0 1 1 1, γ 67 = 0 0 0 0 1 1 1 1, γ 68 = 0 0 1 0 1 1 1 1, γ 69 = 0 0 1 1 1 1 1 1, γ 70 = 0 1 1 0 1 1 1 1, γ 71 = 0 1 1 1 1 1 1 1, γ 72 = 1 1 1 0 1 1 1 1, γ 73 = 0 1 2 1 1 1 1 1, γ 74 = 1 1 1 1 1 1 1 1, γ 75 = 1 1 2 1 1 1 1 1, γ 76 = 0 1 2 1 2 1 1 1, γ 77 = 1 2 2 1 1 1 1 1, γ 78 = 1 1 2 1 2 1 1 1, γ 79 = 0 1 2 1 2 2 1 1, γ 80 = 1 2 2 1 2 1 1 1, γ 81 = 1 1 2 1 2 2 1 1, γ 83 = 1 2 3 1 2 1 1 1, γ 84 = 1 2 2 1 2 2 1 1, γ 86 = 1 2 3 2 2 1 1 1, γ 87 = 1 2 3 1 2 2 1 1, γ 89 = 1 2 3 2 2 2 1 1, γ 90 = 1 2 3 1 3 2 1 1, γ 92 = 1 2 3 2 3 2 1 1, γ 95 = 1 2 4 2 3 2 1 1, γ 98 = 1 3 4 2 3 2 1 1, γ 101 = 2 3 4 2 3 2 1 1, γ 7 = 0 0 0 0 0 0 1 0, γ 39 = 0 0 0 0 0 1 1 0, γ 40 = 0 0 0 0 1 1 1 0, γ 41 = 0 0 1 0 1 1 1 0, γ 42 = 0 0 1 1 1 1 1 0, γ 43 = 0 1 1 0 1 1 1 0, γ 44 = 0 1 1 1 1 1 1 0, γ 45 = 1 1 1 0 1 1 1 0, γ 46 = 0 1 2 1 1 1 1 0, γ 47 = 1 1 1 1 1 1 1 0, γ 48 = 1 1 2 1 1 1 1 0, γ 49 = 0 1 2 1 2 1 1 0, γ 50 = 1 2 2 1 1 1 1 0, γ 51 = 1 1 2 1 2 1 1 0, γ 52 = 0 1 2 1 2 2 1 0, γ 53 = 1 2 2 1 2 1 1 0, γ 54 = 1 1 2 1 2 2 1 0, γ 55 = 1 2 3 1 2 1 1 0, γ 56 = 1 2 2 1 2 2 1 0, γ 57 = 1 2 3 2 2 1 1 0, γ 58 = 1 2 3 1 2 2 1 0, γ 59 = 1 2 3 2 2 2 1 0, γ 60 = 1 2 3 1 3 2 1 0, γ 61 = 1 2 3 2 3 2 1 0, γ 62 = 1 2 4 2 3 2 1 0, γ 63 = 1 3 4 2 3 2 1 0, γ 64 = 2 3 4 2 3 2 1 0 }, where a 1 a 3 a 4 a 5 a 6 a 7 a 8 a 2 stands for the root γ = 8 a iα i Φ +. We fix the orientation (or sign) of y γ as in the following: For γ i Φ + (i = 65,..., 120), we put y i = y γi and fix the orientation by the following order; y 65 = [y 7, y 8 ], y 66 = [y 6, y 65 ], y 67 = [y 5, y 66 ], y 68 = [y 4, y 67 ], y 69 = [y 2, y 68 ], y 70 = [y 3, y 68 ], y 71 = [y 2, y 70 ], y 72 = [y 1, y 70 ], y 73 = [y 4, y 71 ], y 74 = [y 1, y 71 ], y 75 = [y 1, y 73 ], y 76 = [y 5, y 73 ], y 77 = [y 3, y 75 ], y 78 = [y 5, y 75 ], y 79 = [y 6, y 76 ], y 80 = [y 5, y 77 ], y 81 = [y 6, y 78 ], y 82 = [y 7, y 79 ], y 83 = [y 7, y 80 ], y 84 = [y 6, y 80 ], y 85 = [y 1, y 82 ], y 86 = [y 2, y 83 ], y 87 = [y 6, y 83 ], y 88 = [y 3, y 85 ], y 89 = [y 2, y 87 ], y 90 = [y 5, y 87 ], y 91 = [y 4, y 88 ], y 92 = [y 7, y 90 ], y 93 = [y 7, y 91 ], y 94 = [y 5, y 91 ], y 95 = [y 4, y 92 ], y 96 = [y 2, y 94 ], y 97 = [y 6, y 94 ], y 98 = [y 3, y 95 ], y 99 = [y 4, y 96 ], y 100 = [y 2, y 97 ], y 101 = [y 1, y 98 ], y 102 = [y 3, y 99 ], y 103 = [y 4, y 100 ], y 104 = [y 1, y 102 ], y 105 = [y 3, y 103 ], y 106 = [y 5, y 103 ], y 107 = [y 1, y 105 ], y 108 = [y 5, y 105 ], y 109 = [y 4, y 108 ], y 110 = [y 1, y 108 ], 20

y 111 = [y 2, y 109 ], y 112 = [y 4, y 110 ], y 113 = [y 2, y 112 ], y 114 = [y 3, y 112 ], y 115 = [y 2, y 114 ], y 116 = [y 4, y 115 ], y 117 = [y 5, y 116 ], y 118 = [y 6, y 117 ], y 119 = [y 7, y 118 ], y 120 = [y 8, y 119 ]. Then, by the repeated application of Jacobi identities, we obtain y 120 = [ y 118, y 65 ] = [y 117, y 66 ] = [ y 116, y 67 ] = [y 115, y 68 ] = [y 114, y 69 ] = [ y 113, y 70 ] = [y 112, y 71 ] = [y 111, y 72 ] = [y 110, y 73 ] = [y 109, y 74 ] = [y 108, y 75 ] = [y 107, y 76 ] = [y 106, y 77 ] = [y 105, y 78 ] = [y 104, y 79 ] = [y 103, y 80 ] = [y 102, y 81 ] = [y 100, y 83 ] = [y 99, y 84 ] = [y 97, y 86 ] = [y 96, y 87 ] = [y 94, y 89 ] = [y 93, y 90 ] = [y 91, y 92 ] = [y 88, y 95 ] = [y 85, y 98 ] = [y 82, y 101 ] y 119 = [ y 118, y 7 ] = [y 117, y 39 ] = [ y 116, y 40 ] = [y 115, y 41 ] = [y 114, y 42 ] = [ y 113, y 43 ] = [y 112, y 44 ] = [y 111, y 45 ] = [y 110, y 46 ] = [y 109, y 47 ] = [y 108, y 48 ] = [y 107, y 49 ] = [y 106, y 50 ] = [y 105, y 51 ] = [y 104, y 52 ] = [y 103, y 53 ] = [y 102, y 54 ] = [y 100, y 55 ] = [y 99, y 56 ] = [y 97, y 57 ] = [y 96, y 58 ] = [y 94, y 59 ] = [y 93, y 60 ] = [y 91, y 61 ] = [y 88, y 62 ] = [y 85, y 63 ] = [y 82, y 64 ] Thus, putting W 1 = y 120, W 2 = y 119, Z 1 = y 118, Z 2 = y 117, Z 3 = y 116, Z 4 = y 115, Z 5 = y 114, Z 6 = y 113, Z 7 = y 112, Z 8 = y 111, Z 9 = y 110, Z 10 = y 109, Z 11 = y 108, Z 12 = y 107, Z 13 = y 106, Z 14 = y 105, Z 15 = y 104, Z 16 = y 103, Z 17 = y 102, Z 18 = y 100, Z 19 = y 99, Z 20 = y 97, Z 21 = y 96, Z 22 = y 94, Z 23 = y 93, Z 24 = y 91, Z 25 = y 88, Z 26 = y 85, Z 27 = y 82, Y 1 = y 65, Y 2 = y 66, Y 3 = y 67, Y 4 = y 68, Y 5 = y 69, Y 6 = y 70, Y 7 = y 71, Y 8 = y 72, Y 9 = y 73, Y 10 = y 74, Y 11 = y 75, Y 12 = y 76, Y 13 = y 77, Y 14 = y 78, Y 15 = y 79, Y 16 = y 80, Y 17 = y 81, Y 18 = y 83, Y 19 = y 84, Y 20 = y 86, Y 21 = y 87, Y 22 = y 89, Y 23 = y 90, Y 24 = y 92, Y 25 = y 95, Y 26 = y 98, Y 27 = y 101, X 1 = y 7, X 2 = y 39, X 3 = y 40, X 4 = y 41, X 5 = y 42, X 6 = y 43, X 7 = y 44, X 8 = y 45, X 9 = y 46, X 10 = y 47, X 11 = y 48, X 12 = y 49, X 13 = y 50, X 14 = y 51, X 15 = y 52, X 16 = y 53, X 17 = y 54, X 18 = y 55, X 19 = y 56, X 20 = y 57, X 21 = y 58, X 22 = y 59, X 23 = y 60, X 24 = y 61, X 25 = y 62, X 26 = y 63, X 27 = y 64, we obtain the basis {W 1, W 2, Z 1,..., Z 27, Y 1,..., Y 27, X 1,..., X 27 } of m satisfying the following: g 3 = {W 1, W 2 }, g 2 = {Z 1,..., Z 27 }, g 1 = {Y 1,..., Y 27, X 1,..., X 27 } such that [Z i, Y j ] = δ i jw 1, [Z i, X j ] = δ i jw 2 [X i, X j ] = [Y i, Y j ] = 0 (1 i, j 27) 21

Then we calculate [X j, Y k ] for 1 j, k 27 and obtain Z 1 = [X 13, Y 27 ] = [X 16, Y 26 ] = [X 18, Y 25 ] = [X 20, Y 24 ] = [X 22, Y 23 ] = [X 23, Y 22 ] = [X 24, Y 20 ] = [X 25, Y 18 ] = [X 26, Y 16 ] = [X 27, Y 13 ], Z 2 = [X 11, Y 27 ] = [X 14, Y 26 ] = [X 17, Y 25 ] = [X 19, Y 24 ] = [X 21, Y 22 ] = [X 22, Y 21 ] = [X 24, Y 19 ] = [X 25, Y 17 ] = [X 26, Y 14 ] = [X 27, Y 11 ], Z 3 = [X 9, Y 27 ] = [X 12, Y 26 ] = [X 15, Y 25 ] = [X 19, Y 23 ] = [X 20, Y 21 ] = [X 21, Y 20 ] = [X 23, Y 19 ] = [X 25, Y 15 ] = [X 26, Y 12 ] = [X 27, Y 9 ], Z 4 = [X 7, Y 27 ] = [X 10, Y 26 ] = [X 15, Y 24 ] = [X 17, Y 23 ] [X 18, Y 21 ] = [X 21, Y 18 ] = [X 23, Y 17 ] = [X 24, Y 15 ] = [X 26, Y 10 ] = [X 27, Y 7 ], Z 5 = [X 6, Y 27 ] = [X 10, Y 25 ] = [X 12, Y 24 ] = [X 14, Y 23 ] = [X 16, Y 21 ] = [X 21, Y 16 ] = [X 23, Y 14 ] = [X 24, Y 12 ] = [X 25, Y 10 ] = [X 27, Y 6 ], Z 6 = [X 5, Y 27 ] = [X 8, Y 26 ] = [X 15, Y 22 ] = [X 17, Y 20 ] = [X 18, Y 19 ] = [X 19, Y 18 ] = [X 20, Y 17 ] = [X 22, Y 15 ] = [X 26, Y 8 ] = [X 27, Y 5 ], Z 7 = [X 4, Y 27 ] = [X 8, Y 25 ] = [X 12, Y 22 ] = [X 14, Y 20 ] = [X 16, Y 19 ] = [X 19, Y 16 ] = [X 20, Y 14 ] = [X 22, Y 12 ] = [X 25, Y 8 ] = [X 27, Y 4 ], Z 8 = [X 6, Y 26 ] = [X 7, Y 25 ] = [X 9, Y 24 ] = [X 11, Y 23 ] = [X 13, Y 21 ] = [X 21, Y 13 ] = [X 23, Y 11 ] = [X 24, Y 9 ] = [X 25, Y 7 ] = [X 26, Y 6 ], Z 9 = [X 3, Y 27 ] = [X 8, Y 24 ] = [X 10, Y 22 ] = [X 14, Y 18 ] = [X 16, Y 17 ] = [X 17, Y 16 ] = [X 18, Y 14 ] = [X 22, Y 10 ] = [X 24, Y 8 ] = [X 27, Y 3 ], Z 10 = [X 4, Y 26 ] = [X 5, Y 25 ] = [X 9, Y 22 ] = [X 11, Y 20 ] = [X 13, Y 19 ] = [X 19, Y 13 ] = [X 20, Y 11 ] = [X 22, Y 9 ] = [X 25, Y 5 ] = [X 26, Y 4 ], Z 11 = [X 2, Y 27 ] = [X 8, Y 23 ] = [X 10, Y 20 ] = [X 12, Y 18 ] = [X 15, Y 16 ] = [X 16, Y 15 ] = [X 18, Y 12 ] = [X 20, Y 10 ] = [X 23, Y 8 ] = [X 27, Y 2 ], Z 12 = [X 3, Y 26 ] = [X 5, Y 24 ] = [X 7, Y 22 ] = [X 11, Y 18 ] = [X 13, Y 17 ] = [X 17, Y 13 ] = [X 18, Y 11 ] = [X 22, Y 7 ] = [X 24, Y 5 ] = [X 26, Y 3 ], Z 13 = [X 1, Y 27 ] = [X 8, Y 21 ] = [X 10, Y 19 ] = [X 12, Y 17 ] = [X 14, Y 15 ] = [X 15, Y 14 ] = [X 17, Y 12 ] = [X 19, Y 10 ] = [X 21, Y 8 ] = [X 27, Y 1 ], Z 14 = [X 2, Y 26 ] = [X 5, Y 23 ] = [X 7, Y 20 ] = [X 9, Y 18 ] = [X 13, Y 15 ] = [X 15, Y 13 ] = [X 18, Y 9 ] = [X 20, Y 7 ] = [X 23, Y 5 ] = [X 26, Y 2 ], Z 15 = [X 3, Y 25 ] = [X 4, Y 24 ] = [X 6, Y 22 ] = [X 11, Y 16 ] = [X 13, Y 14 ] = [X 14, Y 13 ] = [X 16, Y 11 ] = [X 22, Y 6 ] = [X 24, Y 4 ] = [X 25, Y 3 ], Z 16 = [X 1, Y 26 ] = [X 5, Y 21 ] = [X 7, Y 19 ] = [X 9, Y 17 ] = [X 11, Y 15 ] = [X 15, Y 11 ] = [X 17, Y 9 ] = [X 19, Y 7 ] = [X 21, Y 5 ] = [X 26, Y 1 ], Z 17 = [X 2, Y 25 ] = [X 4, Y 23 ] = [X 6, Y 20 ] = [X 9, Y 16 ] = [X 12, Y 13 ] = [X 13, Y 12 ] = [X 16, Y 9 ] = [X 20, Y 6 ] = [X 23, Y 4 ] = [X 25, Y 2 ], Z 18 = [X 1, Y 25 ] = [X 4, Y 21 ] = [X 6, Y 19 ] = [X 9, Y 14 ] = [X 11, Y 12 ] = [X 12, Y 11 ] = [X 14, Y 9 ] = [X 19, Y 6 ] = [X 21, Y 4 ] = [X 25, Y 1 ], 22