G2- GEOMETRY IN CONTACT GEOMETRY OF SECOND ORDER
KEIZO YAMAGUCHI
1. Introduction
In his famous “Five variables paper [3]”, E. Cartan investigated the (local) contact equiva-
lence 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 involu-
tive, 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
G2.
(A)
∂2z
∂x2
=
1
3
(
∂2z
∂y2
)3
,
∂2z
∂x∂y
=
1
2
(
∂2z
∂y2
)2
.
(B) 9r2 + 12t2(rt− s2) + 32s3 − 36rst = 0,
where
r =
∂2z
∂x2
, s =
∂2z
∂x∂y
, t =
∂2z
∂y2
.
are the classical terminology.
In [13], we observed, for each exceptional simple Lie algebra Xℓ, we could find the overde-
termined system (Aℓ) and the single equation of Goursat type (Bℓ), whose symmetry algebras
are isomporphic with Xℓ and formulated this fact as the G2-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 (Aℓ) 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 (Bℓ) 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. G2-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 ∆ = {α1, . . . , αℓ} 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 Xℓ (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 (Xℓ,∆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 = {α1, αℓ} for Aℓ type and ∆θ = {αθ} for other
types. Here αθ = α2, α1, α2 for Bℓ, Cℓ, Dℓ types respectively and αθ = α2, α1, α2, α1, α8 for G2,
F4, E6, E7, E8 types respectively.
Moreover we have the adjoint (or equivalently coadjoint) representation, which has θ as the
highest weight. The R-space Jg corresponding to (Xℓ,∆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, Jg has the natural contact structure Cg induced from the symplectic
structure as the coadjoint orbit, which corresponds to the contact gradation (Xℓ,∆θ) (cf. [12],
§4). Standard contact manifolds (Jg, Cg) 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])
Aℓ (ℓ > 1)
◦ ◦ ....... ◦ ◦

◦HHHH1 1 1 1
−θ
α1 α2 αℓ−1 αℓ
Bℓ (ℓ > 2)
◦HH
◦
◦ ....... ◦=⇒◦1
2 2 2
−θ
α1
α2 αℓ−1 αℓ
Cℓ (ℓ > 1)
◦=⇒◦ ....... ◦⇐=◦
2 2 1
−θ α1 αℓ−1 αℓ
Dℓ (ℓ > 3)
◦HH
◦
◦ ....... ◦
◦
HH◦1
2 2
1
1
−θ
α1
α2 αℓ−2
αℓ−1
αℓ
E6
◦ ◦ ◦ ◦ ◦
◦
◦
1 2 3 2 1
2
−θ
α1 α3 α4 α5 α6
α2
F4
◦ ◦ ◦=⇒◦ ◦2 3 4 2
−θ α1 α2 α3 α4
E7
◦ ◦ ◦ ◦ ◦ ◦ ◦
◦
2 3 4 3 2 1
2
−θ α1 α3 α4 α5 α6 α7
α2 G2
◦⇐=◦ ◦3 2
−θα1 α2
E8
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦
◦
2 4 6 5 4 3 2
3
−θα1 α3 α4 α5 α6 α7 α8
α2
2.2. G2-geometry. Let (Xℓ,∆θ) be the (standard) contact gradation. Then we have ∆θ =
{αθ} except for Aℓ 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 αG next to αθ such that
the coefficient of αG in the highest root is 3 (see the above diagrams). In the classical cases,
the pair {α1, α3} of simple roots plays the role of αG in BDℓ type. We will consider simple
graded Lie algebras (Xℓ, {αG}) 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 Xℓ. 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 PD manifolds (R(X), D1X , D2X) from (X,D). Our
2
model overdetermined system (Aℓ) will be the PD manifold of second order constructed from
the standard differential system of type m, where m is the Goursat gradation of type Xℓ.
Explicitly we will here consider the following simple graded Lie algebras of depth 3: (G2, {α1}),
(F4, {α2}), (E6, {α4}), (E7, {α3}), (E8, {α7}), (Bℓ, {α1, α3}) (ℓ ≧ 3), (Dℓ, {α1, α3}) (ℓ ≧ 5) and
(D4, {α1, α3, α4}). These graded Lie algebras have the common feature with (G2, {α1}) as fol-
lows: The Goursat gradation m = g−3⊕g−2⊕g−1 satisfies dim g−3 = 2 and dim g−1 = 2dim 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 − αθ} such that the coefficient of αθ in each
β ∈ Φ+2 is 1 and Φ+1 consists of roots θ − β, θ − αθ − 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 dimW = 2 and dimV = s. Here s = 1, 6, 9, 15, 27, 2ℓ − 4 or 2ℓ − 5 corrresponding to
Xℓ = G2, F4, E6, E7, E8, Bℓ or Dℓ.
Thus let (X,D) be a regular differential system of type m, where m is the Goursat gradation
of type Xℓ. Then (X, ∂D) is a regular differential system of type c
1(s, 2), where c1(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
{ϖ1, ϖ2, ω1, . . . , ωs, π11, . . . , πs1, π12, . . . , πs2} around x ∈ X such that
∂D = {ϖ1 = ϖ2 = 0}, D = {ϖ1 = ϖ2 = ω1 = · · · = ωs = 0},
and {
dϖ1 ≡ π11 ∧ ω1 + · · ·+ πs1 ∧ ωs (mod ϖ1, ϖ2)
dϖ2 ≡ π12 ∧ ω1 + · · ·+ πs2 ∧ ωs (mod ϖ1, ϖ2)
Thus (X,D) satisfies the conditions (X.1) to (X.3) in §4.3 [14]. Hence we can construct the
PD manifold (R(X);D1X , D
2
X) as follows: Let us consider the collection R(X) of hyperplanes
v in each tangent space Tx(X) at x ∈ X which contains the fibre ∂D(x) of the derived system
∂D of D.
R(X) =
∪
x∈X
Rx ⊂ J(X, 3s+ 1),
Rx = {v ∈ Gr(Tx(X), 3s+ 1) | v ⊃ ∂D(x)} ∼= P(Tx(X)/∂D(x)) = P1.
Moreover D1X is the canonical system obtained by the Grassmaniann construction and D
2
X is
the lift of D. In fact, D1X and D
2
X are given by
D1X(v) = ρ
−1
∗ (v) ⊃ D2X(v) = 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 c1(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 = dimW and s = dimV . Let (Y,C) be a regular differential system of type c1(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 c1(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 (Mg, Dg) of type (Xℓ, {αG}), R(X) can be iden-
tified with the model space (Rg, Eg) of type (Xℓ, {αθ, αG}) as follows (here, we understand αG
denotes two simple roots α1 and α3 in case of BDℓ types and three simple roots α1,α3 and α4
in case of D4): Let (Jg, Cg) be the standard contact manifold of type (Xℓ, {αθ}). Then we have
the double fibration;
Rg
πc−−−→ Jg
πg
y
Mg
Here (Xℓ, {αθ, αG}) 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 (Xℓ, {αG}), we can check that Φ̌+5 = {θ}, Φ̌+4 = {θ − αθ}, Φ̌+3 = Φ+2 , Φ̌+2 consists of roots
θ − β for each β ∈ Φ̌+3 and Φ̌+1 consists of roots αθ and θ − αθ − β for each β ∈ Φ̌+3 (see
§6.2 [14] for BDℓ-type). Thus we see that ∂(3)Eg = (πc)−1∗ (Cg), ∂(2)Eg = (πg)−1∗ (∂Dg) and
∂Eg = (πg)
−1
∗ (Dg). We put D
1 = ∂(3)Eg and D
2 = ∂Eg. Then (Rg;D
1, D2) is a PD manifold of
second order. In fact, we have an isomorphism of (Rg;D
1, D2) onto (R(Mg);D
1
Mg
, D2Mg) by the
Realization Lemma for (Rg, D
1, πg,Mg) and an embedding of Rg into L(Jg) by the Realization
Lemma for (Rg, D
2, πc, Jg). Thus Rg is identified with a R-space orbit in L(Jg). Moreover,
putting C∗ = ∂(3)Eg and N = ∂
(2)Eg, (Rg;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 Xℓ. 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 PD manifold (R(X);D1X , D
2
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 (Bℓ)
as in §5 (see Remark 6.2 (1) and §8.3 [15] for BDℓ-type).
2.3. Goursat gradations of Exceptional simple Lie algebras. Let m = g−3⊕g−2⊕g−1 be
the Goursat gradation of type Xℓ, where Xℓ 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 − αθ}, Φ+2 = {β1, . . . , βs}, Φ+1 = {αθ + γ1, . . . , αθ + γs, γ1, . . . , γs}.
where θ− (2αθ +3αG), βi − (αθ +2αG) and γi −αG are spanned by simple roots other than αθ
and αG of Xℓ such that θ = βi + αθ + γi (i = 1, . . . , s).
Then we will calculate the structure of m explicitly by use of the Chevalley basis of Xℓ. By
adjusting the Chevalley basis (especially, for E6, E7, E8, by changing the orientation of Chevalley
basis) suitably (see §4 for detail), we obtain the basis {W1,W2, Z1, . . . , Zs, Y1, . . . , Ys, X1, . . . , Xs}
of m satisfying the following:
g−3 = ⟨{W1,W2}⟩, g−2 = ⟨{Z1, . . . , Zs}⟩, g−1 = ⟨{Y1, . . . , Ys, X1, . . . , Xs}⟩
such that
[Zi, Yj] = δ
i
jW1, [Zi, Xj] = δ
i
jW2 [Xi, Xj] = [Yi, Yj] = 0 (1 ≦ i, j ≦ s)
In these basis, we calculate [Xj, Yk] for 1 ≦ j, k ≦ s in §4.
4
3. Classical Cases (BDℓ-type)
The structure of the Goursat gradation m of type BDℓ,
m = g−3 ⊕ g−2 ⊕ g−1
is given by the following brackets among the basis {W1,W2, Z1, . . . , Zp+1, Y1, . . . , Yp+1, X1, . . . , Xp+1}
of m ;
g−3 = ⟨{W1,W2}⟩, g−2 = ⟨{Z1, . . . , Zp+1}⟩, g−1 = ⟨{Y1, . . . , Yp+1, X1, . . . , Xp+1}⟩
such that
(3.1) [Zi, Yj] = δ
i
jW1, [Zi, Xj] = δ
i
jW2, [Xk1 , Yk2 ] = δ
k1
k2
Z1, [X1, Yk] = [Xk, Y1] = Zk,
[Xi, Xj] = [Yi, Yj] = 0 (1 ≦ i, j ≦ p+ 1, 2 ≦ k, k1, k2 ≦ 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 = C3p+5
describes the standard differential system of type m
D = {ϖ1 = ϖ2 = ω1 = · · · = ωp+1 = 0 },
where 
ϖ1 = dw1 − (z1 + 12
p+1∑
k=2
xkyk) dy1 −
p+1∑
k=2
{zk + 12(xky1 + x1yk)} dyk,
ϖ2 = dw2 − (z1 − 12
p+1∑
k=2
xkyk) dx1 −
p+1∑
k=2
{zk − 12(xky1 + x1yk)} dxk,
ω1 = dz1 +
1
2
p+1∑
k=2
(yk dxk − xk dyk),
ωk = dzk +
1
2
(y1 dxk − xk dy1) + 12(yk dx1 − x1 dyk), (2 ≦ k ≦ p+ 1).
and (w1, w2, z1, . . . , zp+1, y1, . . . , yp+1, x1, . . . , xp+1) is a coordinate system of X = C3p+5. In fact
we have
(3.2)

dϖ1 = dy1 ∧ ω1 + · · ·+ dyp+1 ∧ ωp+1,
dϖ2 = dx1 ∧ ω1 + · · ·+ dxp+1 ∧ ωp+1,
dω1 = dy2 ∧ dx2 + · · ·+ dyp+1 ∧ dxp+1,
dωk = dy1 ∧ dxk + dyk ∧ dx1 (2 ≦ k ≦ p+ 1),
which is the dual of (3.1). In particular, we have
∂D = {ϖ1 = ϖ2 = 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 (BDℓ, {α1, α3}). As in §2.2, (R(X);D1X , D2X) is constructed as follows;
R = R(X) is the collection of hyperplanes v in each tangent space Tx(X) at x ∈ X which
contains the fobre ∂D of D.
R(X) =
∪
x∈X
Rx ⊂ J(X, 3p+ 4),
Rx = {v ∈ Gr(Tx(X), 3p+ 4) | v ⊃ ∂D(x)} ∼= P1,
5
Moreover D1 is the canonical system obtained by the Grassmaniann construction and D2 is the
lift of D. Precisely, D1 and D2 are given by
D1(v) = ρ−1∗ (v) ⊃ D2(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 = ϖ1 + λϖ2, where
D1 = {ϖ = 0 } and ∂D = {ϖ1 = ϖ2 = 0}.
Here (w1, w2, z1, . . . , zp+1, y1, . . . , yp+1, x1, . . . , xp+1, λ) constitute a coordinate system on R(X).
Then we have
dϖ = (dy1 + λdx1) ∧ ω1 + · · ·+ (dyp+1 + λdxp+1) ∧ ωp+1 + dλ ∧ϖ2,
Ch (D1) = { ϖ = ϖ2 = ω1 = · · · = ωp+1 = dy1 + λdx1 = · · · = dyp+1 + λdxp+1 = dλ = 0 },
D2 = { ϖ = ϖ2 = ω1 = · · · = ωp+1 = 0 } and ∂D2 = { ϖ = ϖ2 = 0 }.
Thus (R(X);D1, D2) is a PD-manifold of seconod rder. Now we calculate
ϖ = ϖ1 + λϖ2
= dw1 − (z1 + 12
p+1∑
k=2
xkyk) dy1 −
p+1∑
k=2
{zk + 12(xky1 + x1yk)} dyk,
+ λ{dw2 − (z1 − 12
p+1∑
k=2
xkyk) dx1 −
p+1∑
k=2
{zk − 12(xky1 + x1yk)} dxk},
= dw1 + λdw2 − (z1 + 12
p+1∑
k=2
xkyk) (dy1 + λdx1)−
p+1∑
k=2
{zk + 12(xky1 + x1yk)} (dyk + λdxk)
+ λ{(
p+1∑
k=2
xkyk)dx1 +
p+1∑
k=2
(xky1 + x1yk)dxk}
= d(w1 + λw2)− (z1 + 12
p+1∑
k=2
xkyk) d(y1 + λx1)−
p+1∑
k=2
{zk + 12(xky1 + x1yk)} d(yk + λxk)
− {w2 − (z1 + 12
p+1∑
k=2
xkyk)x1 −
p+1∑
k=2
{zk + 12(xky1 + x1yk)}xk}dλ
+ λ{(
p+1∑
k=2
xkyk)dx1 +
p+1∑
k=2
(xky1 + x1yk)dxk}
Moreover we calculate
λ{(
p+1∑
k=2
xkyk)dx1 +
p+1∑
k=2
(xky1 + x1yk)dxk} =
p+1∑
k=2
{λykd(x1 xk) + 12λy1d(xk)
2}
=
p+1∑
k=2
{d(λx1xkyk + 12λx
2
ky1)− x1xkd(λyk)− 12x
2
kd(λy1)}
6
=
p+1∑
k=2
{d(λx1xkyk + 12λx
2
ky1)− (x1xkyk + 12x
2
ky1)dλ− λx1xkdyk − 12λx
2
kdy1}
=
p+1∑
k=2
{d(λx1xkyk + 12λx
2
ky1)− (x1xkyk + 12x
2
ky1)dλ
− λx1xkd(yk + λxk)− 12λx
2
kd(y1 + λx1) + λx1xkd(λxk) +
1
2
λx2kd(λx1)}
=
p+1∑
k=2
{d(λx1xkyk + 12λx
2
ky1 +
1
2
λ2x1x
2
k)− (x1xkyk + 12x
2
ky1 − 12λx1x
2
k)dλ
− λx1xkd(yk + λxk)− 12λx
2
kd(y1 + λx1)}
Thus we obtain
ϖ = dZ − P1dX1 −
p+2∑
k=2
PkdXk
where 
Z = w1 + λw2 + λ(x1
∑p+1
k=2 xkyk +
1
2
∑p+1
k=2 x
2
ky1) +
1
2
λ2 x1
∑p+1
k=2 x
2
k,
P1 = w2 −
∑p+1
i=1 xizi −
1
2
λx1
∑p+1
k=2 x
2
k,
P2 = z1 +
1
2
∑p+1
k=2 xkyk +
1
2
λ
∑p+1
k=2 x
2
k,
Pk+1 = zk +
1
2
(xky1 + x1yk) + λx1xk, (2 ≦ k ≦ p+ 1),
X1 = λ,
Xi+1 = yi + λxi (1 ≦ i ≦ p+ 1).
Thus
D1 = { dZ −
p+2∑
i=1
Pi dXi = 0 },
and (X1, . . . , Xp+2, Z, P1, . . . , Pp+2) constitutes a canonical coordinate system on J = R(X)/Ch (D
1).
Putting xi = ai, we solve
λ = X1, yi = Xi+1 − aiX1 (1 ≦ i ≦ p+ 1),
z1 = P2 − 12
∑p+1
k=2 akXk+1,
zk = Pk+1 − 12(akX2 + a1Xk+1) (2 ≦ k ≦ p+ 1),
w2 = P1 + a1(P2 − 12
∑p+1
k=2 akXk+1)
+
∑p+1
k=2 ak{Pk+1 −
1
2
(akX2 + a1Xk+1)}+ 12a1
∑p+1
p=2 a
2
k X1
= P1 +
∑p+1
k=1 akPk+1 +
1
2
a1
∑p+1
k=2 a
2
k X1 − 12
∑p+1
k=2 a
2
k X2 −
∑p+1
k=2 a1ak Xk+1.
Then we calculate
(3.3)

ω1 = dP2 +
1
2
p+1∑
k=2
a2k dX1 −
p+1∑
k=2
ak dXk+1,
ωk = dPk+1 + a1ak dX1 − ak dX2 − a1 dXk+1 (k = 2, . . . , p+ 1),
ϖ2 = dP1 +
p+1∑
k=2
akdPk+1 +
1
2
p+1∑
k=2
a1a
2
k dX1 − 12
p+1∑
k=2
a2k dX2 −
p+1∑
k=2
a1ak dXk+1,
=
p+1∑
k=1
akωk + dP1 −
p+1∑
k=2
a1a
2
k dX1 +
1
2
p+1∑
k=2
a2k dX2 +
p+1∑
k=2
a1ak dXk+1.
7
This implies R(X) is given by the following 1
2
(p+ 1)(p+ 2) + 1 equations;
P22 = 0, Pij = δijP33 (3 ≦ i, j ≦ p+ 2),
P11 = P33
p+1∑
k=2
P 22,k+1, P12 = −12
p+1∑
k=2
P 22,k+1, P1,k+1 = −P33P2,k+1 (2 ≦ k ≦ p+ 1),
in terms of the canonical coordinate (X1, . . . , Xp+2, Z, P1, . . . , Pp+2, P11, . . . , Pp+2,p+2) of L(J).
4. Exceptional Cases
Let m = g−3⊕g−2⊕g−1 be the Goursat gradation of typeXℓ, whereXℓ is one of the excetional
simple Lie algebra. As in §2.3, we choose the basis {W1,W2, Z1, . . . , Zs, Y1, . . . , Ys, X1, . . . , Xs}
of m satisfying the following:
g−3 = ⟨{W1,W2}⟩, g−2 = ⟨{Z1, . . . , Zs}⟩, g−1 = ⟨{Y1, . . . , Ys, X1, . . . , Xs}⟩
such that
[Zi, Yj] = δ
i
jW1, [Zi, Xj] = δ
i
jW2 [Xi, Xj] = [Yi, Yj] = 0 (1 ≦ i, j ≦ s)
Utilizing the bilinear forms fi(x1, . . . , xs, y1, . . . , ys) for i = 1, . . . , s, which describes the brackets
[Xj, Yk] (see the following subsections), we can describe the standard differential system of type
m as follows: Let (w1, w2, z1, . . . , zs, y1, . . . , ys, x1, . . . , xs)be the linear coordinate of X = m
given by the above basis of m. Then (X,D) on X = C3s+2 describes the standard differential
system of type m
D = {ϖ1 = ϖ2 = ω1 = · · · = ωs = 0 },
where
ϖ1 = dw1 −
s∑
i=1
(zi +
1
2
fi) dyi, ϖ2 = dw2 −
s∑
i=1
(zi − 12fi) dxi
and
ωi = dzi − 12{fi(xk, dyk)− fi(dxk, yk)} (i = 1, . . . , s).
In fact we have
dωi = −fi(dxk ∧ dyk) (i = 1, . . . , s)
and we can calculate in each case
dϖ1 = dy1 ∧ ω1 + dy2 ∧ ω2 + · · ·+ dys ∧ ωs,
dϖ2 = dx1 ∧ ω1 + dx2 ∧ ω2 + · · ·+ dxs ∧ ωs,
which describes the structure of m. In particular, we have
∂D = {ϖ1 = ϖ2 = 0}
By the First Reduction Theorem, as in §2.2, our model overdetermined system (R(X);D1X , D2X)
is constructed from (X,D) as follows:
R(X) =
∪
x∈X
Rx ⊂ J(X, 3s+ 1),
Rx = {v ∈ Gr(Tx(X), 3s+ 1) | v ⊃ ∂D(x)} ∼= P1,
Moreover D1X and D
2
X are given by
D1X(v) = ρ
−1
∗ (v) ⊃ D2X(v) = 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 = ϖ1 + λϖ2, where
D1 = {ϖ = 0 } and ∂D = {ϖ1 = ϖ2 = 0}.
Here (w1, w2, z1, . . . , zs, y1, . . . , ys, x1, . . . , xs, λ) constitutes a coordinate system on R(X).
Now we put
s∑
i=1
xi fi = 2
s∑
i=1
yi gi
where gi(x1, . . . , xs) = gi(xk) =
1
2
fi(xk, xk).
Then we can check in each case
s∑
i=1
fi dxi =
s∑
i=1
yi dgi,
s∑
i=1
gi dxi = dg, 3g =
s∑
i=1
xi gi,
where g is a cubic polynomial in x1, . . . , xs.
Now, for ϖ = ϖ1 + λϖ2, we calculate
dϖ = (dy1 + λdx1) ∧ ω1 + · · ·+ (dys + λdxs) ∧ ωs + dλ ∧ϖ2,
ϖ = ϖ1 + λϖ2
= dw1 −
s∑
i=1
(zi +
1
2
fi)dyi + λ{dw2 −
s∑
i=1
(zi − 12fi)dxi}
= dw1 + λdw2 −
s∑
i=1
(zi +
1
2
fi)(dyi + λdxi) + λ
s∑
i=1
fidxi
= d(w1 + λw2)− {w2 −
s∑
i=1
(xi zi +
1
2
xi fi)}dλ−
s∑
i=1
(zi +
1
2
fi)d(yi + λxi) + λ
s∑
i=1
fidxi
Moreover we calculate
λ
s∑
i=1
fidxi = λ
s∑
i=1
yidgi = λ
s∑
i=1
{d(yi gi)− gi dyi}
= d(λ
s∑
i=1
yi gi)− (
s∑
i=1
yi gi)dλ− λ
s∑
i=1
gi dyi
= d(λ
s∑
i=1
yi gi)− (12
s∑
i=1
xi fi)dλ− λ
s∑
i=1
gi dyi
λ
s∑
i=1
gidyi = λ
s∑
i=1
gi d(yi + λxi)− λ(
s∑
i=1
xi gi)dλ− λ2
s∑
i=1
gidxi
= λ
s∑
i=1
gi d(yi + λxi)− 3gλdλ− λ2dg
= λ
s∑
i=1
gi d(yi + λxi)− λgdλ− d(λ2g)
Thus we obtain
ϖ = dZ −
s+1∑
i=1
PidXi
9
where 
Z = w1 + λw2 + λ
∑s
i=1 yi gi + λ
2 g
P1 = w2 −
∑s
i=1 xi zi − λ g,
Pi+1 = zi +
1
2
fi + λ gi (i = 1, . . . , s),
X1 = λ,
Xi+1 = yi + λxi (i = 1, . . . , s),
Hence we have
D1 = { dZ − P1 dX1 − · · · − Ps+1 dXs+1 = 0 },
and (X1, . . . , Xs+1, Z, P1, . . . , Ps+1) constitutes a canonical coordinate system on J = R(X)/Ch (D
1).
Putting xi = ai (i = 1, . . . , s) , we solve
λ = X1, yi = Xi+1 − aiX1 (i = 1, . . . , s),
From fi(ak, Xk+1) = fi(ak, yk + λak) = fi(ak, yk) + 2λgi(ak), we have
zi = Pi+1 − 12fi − λgi = Pi+1 −
1
2
fi(ak, Xk+1) (i = 1, . . . , s)
From
∑s
k=1 xk(
1
2
fk + λgk) =
∑s
k=1(yk + λxk)gk, we have
w2 = P1 +
s∑
k=1
akPk+1 + ĝX1 −
s∑
k=1
ĝkXk+1
where ĝ = g(a1, . . . , as), and ĝk = gk(a1, . . . , as) (k = 1, . . . , s).
Moreover we calculate
ωi = dzi − 12{fi(xk, dyk)− fi(dxk, yk)}
= d{Pi+1 − 12fi(ak, Xk+1)} −
1
2
{fi(ak, d(Xk+1 − akX1))− fi(dak, Xk+1 − akX1)}
= dPi+1 − fi(ak, dXk+1) + 12fi(ak, ak)dX1
= dPi+1 − fi(ak, dXk+1) + gi(ak, ak)dX1 (i = 1, . . . , s)
ϖ2 = dw2 −
s∑
i=1
(zi − 12fi)dxi
= d(P1 +
s∑
k=1
akPk+1 + ĝX1 −
s∑
k=1
ĝkXk+1)−
s∑
k=1
Pk+1dak +
s∑
i=1
fidxi +X1dĝ
= dP1 +
s∑
k=1
akdPk+1 + ĝdX1 −
s∑
k=1
ĝkdXk+1
= dP1 + (
s∑
k=1
akωk + 2
s∑
k=1
ĝkdXk+1 − 3ĝdX1) + ĝdX1 −
s∑
k=1
ĝkdXk+1
=
s∑
k=1
akωk + dP1 − 2ĝdX1 +
s∑
k=1
ĝkdXk+1.
10
Thus we obtain
(4.1)

ωi = dPi+1 − fi(ak, dXk+1) + ĝidX1 (i = 1, . . . , s),
ϖ2 =
s∑
k=1
akωk + dP1 − 2ĝdX1 +
s∑
k=1
ĝkdXk+1,
= dP1 +
s∑
k=1
akdPk+1 + ĝdX1 −
s∑
k=1
ĝkdXk+1.
We will utilize the above formulae to describe our model system for each Xℓ.
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 {xα(α ∈ Φ);hi(1 ≦ i ≦ ℓ)} of
the Exceptional simple Lie algebra of type Xℓ and put yβ = x−β for β ∈ Φ+ ( cf. Chapter VII
[5]). We will describe the structure of the Goursat gradation m in terms of {yβ}β∈Φ+ . Moreover
we will regard simple Lie algebras E6 and E7 as regular subalgebras of E8 and utilize the root
space decomposition of E8 to describe the structure of the Goursat gradation of E6 and E7.
4.1. Goursat gradation and model system of type F4. Let m be the Goursat gradation
of type F4. For (F4, {α2}), we have
Φ+3 = {β24 = 2342, β23 = 1342},
Φ+2 = {β22 = 1242, β21 = 1232, β20 = 1222, β19 = 1231, β18 = 1221, β16 = 1220},
Φ+1 = {β5 = 1100, β8 = 1110, β11 = 1120, β12 = 1111, β15 = 1121, β17 = 1122,
β2 = 0100, β6 = 0110, β9 = 0120, β10 = 0111, β13 = 0121, β14 = 0122}
where a1a2a3a4 stands for the root β =
∑4
i=1 aiαi ∈ Φ+.
We fix the orientation (or sign) of yβ as in the following: First we choose the orientation of
yαi for simple roots by fixing the root vectors yi = yαi ∈ g−αi for i = 1, 2, 3, 4. For βi ∈ Φ+(i =
1, . . . , 24), we put yi = yβi and fix the orientation by the following order;
y5 = [y1, y2], y6 = [y2, y3], y8 = [y3, y5], 2y9 = [y3, y6], y10 = [y4, y6],
y11 = [y1, y9], y12 = [y4, y8], y13 = [y3, y10], 2y14 = [y4, y13], y15 = [y4, y11],
y16 = [y2, y11], y17 = [y2, y14], y18 = [y4, y16], y19 = [y3, y18], y20 = [y2, y17],
y21 = [y4, y19], 2y22 = [y3, y21], y23 = [y2, y22], y24 = [y1, y23].
Then, for example, we calculate
[y1, y6] = [y1, [y2, y3]] = [[y1, y2], y3] = [y5, y3] = −y8.
In this way, by the repeated application of Jacobi identities, we obtain
2y24 = [−2y22, y5] = [y21, y8] = [2y20, y11] = [y19,−y12] = [−y18, y15] = [2y16, y17].
and
2y23 = [−2y22, y2] = [y21,−y6] = [2y20, y9] = [y19, y10] = [−y18, y13] = [2y16, y14].
Thus, putting
W1 = 2y24, W2 = 2y23,
Z1 = −2y22, Z2 = y21, Z3 = 2y20, Z4 = y19, Z5 = −y18, Z6 = 2y16,
Y1 = y5, Y2 = y8, Y3 = y11, Y4 = −y12, Y5 = y15, Y6 = y17,
X1 = y2, X2 = −y6, X3 = y9, X4 = y10, X5 = y13, X6 = y14
11
we obtain the basis {W1,W2, Z1, . . . , Z6, Y1, . . . , Y6, X1, . . . , X6} of m satisfying the following:
g−3 = ⟨{W1,W2}⟩, g−2 = ⟨{Z1, . . . , Z6}⟩, g−1 = ⟨{Y1, . . . , Y6, X1, . . . , X6}⟩
such that
[Zi, Yj] = δ
i
jW1, [Zi, Xj] = δ
i
jW2 [Xi, Xj] = [Yi, Yj] = 0 (1 ≦ i, j ≦ 6)
Then we calculate [Xj, Yk] for 1 ≦ j, k ≦ 6 and obtain
Z1 = 2[X3, Y6] = −[X5, Y5] = 2[X6, Y3],
Z2 = [X2, Y6] = [X4, Y5] = [X5, Y4] = [X6, Y2],
Z3 = 2[X1, Y6] = [X4, Y4] = 2[X6, Y1],
Z4 = [X2, Y5] = [X3, Y4] = [X4, Y3] = [X5, Y2],
Z5 = −[X1, Y5] = [X2, Y4] = [X4, Y2] = −[X5, Y1],
Z6 = 2[X1, Y3] = [X2, Y2] = 2[X3, Y1]
Here we define the bilinear forms fi(x1, . . . , x6, y1, . . . , y6)(i = 1, . . . , 6) as follows;
f1 = 2x3 y6 − x5 y5 + 2x6 y3,
f2 = x2 y6 + x4 y5 + x5 y4 + x6 y2,
f3 = 2x1 y6 + x4 y4 + 2x6 y1,
f4 = x2 y5 + x3 y4 + x4 y3 + x5 y2,
f5 = −x1 y5 + x2 y4 + x4 y2 − x5 y1,
f6 = 2x1 y3 + x2 y2 + 2x3 y1,
Moreover we put
6∑
i=1
xi fi = 2
6∑
i=1
yi gi
where the quadratic forms gi(x1, . . . , x6)(i = 1, . . . , 6) are given by{
g1 = 2x3 x6 − 12x
2
5, g2 = x2 x6 + x4 x5, g3 = 2x1 x6 +
1
2
x24,
g4 = x2 x5 + x3 x4, g5 = −x1 x5 + x2 x4, g6 = 2x1 x3 + 12x
2
2,
Then we have
6∑
i=1
fi dxi =
6∑
i=1
yi dgi,
6∑
i=1
gi dxi = dg, 3g =
6∑
i=1
xi gi,
where g(x1, . . . , x6) is the cubic form given by
g = 2x1 x3 x6 + x2 x4 x5 +
1
2
(−x1 x25 + x22x6 + x3x24).
12
Thus, by (4.1), we obtain
ω1 = dP2 + (2a3a6 − 12a
2
5)dX1 − 2a6dX4 + a5dX6 − 2a3dX7,
ω2 = dP3 + (a2a6 + a4a5)dX1 − a6dX3 − a5dX5 − a4dX6 − a2dX7,
ω3 = dP4 + (2a1a6 +
1
2
a24)dX1 − 2a6dX2 − a4dX5 − 2a1dX7,
ω4 = dP5 + (a2a5 + a3a4)dX1 − a5dX3 − a4dX4 − a3dX5 − a2dX6,
ω5 = dP6 + (−a1a5 + a2a4)dX1 + a5dX2 − a4dX3 − a2dX5 + a1dX6,
ω6 = dP7 + (2a1a3 +
1
2
a22)dX1 − 2a3dX2 − a2dX3 − 2a1dX4,
ϖ2 = a1ω1 + a2ω2 + · · ·+ a6ω6+
dP1 − (4a1a3a6 + 2a2a4a5 − a1a25 + a22a6 + a3a24)dX1
+ (2a3a6 − 12a
2
5)dX2 + (a2a6 + a4a5)dX3 + (2a1a6 +
1
2
a24)dX4
+ (a2a5 + a3a4)dX5 + (−a1a5 + a2a4)dX6 + (2a1a3 + 12a
2
2)dX7
This implies that our model system R(X) of type F4 is given by the following 22 equations;
−P66 = 12P47 (= a1), P37 = P56 (= a2), P55 =
1
2
P27 (= a3),
P36 = P45 (= a4), P35 = −P26 (= a5), P33 = 12P24 (= a6),
P22 = P23 = P25 = P34 = P44 = P46 = P57 = P67 = P77 = 0,
P17 = 2P55P66 − 12P
2
37, P16 = −P35P66 − P36P37, P15 = −P35P37 − P36P55,
P14 = 2P33P66 − 12P
2
36, P13 = −P33P37 − P35P36, P12 = 12P
2
35 − 2P33P55,
P11 = −4P33P55P66 + 2P35P36P37 + P 235P66 + P33P 237 + P 236P55
in terms of the canonical coordinate (X1, . . . , X7, Z, P1, . . . , P7, P11, . . . , P77) of L(J).
4.2. Goursat gradation and model system of type E6. In the following subsections , let
us fix the root space decomposition of Simple Lie algebra E8 and regard E6 and E7 as the
regular subalgebras of E8.
Let m be the Goursat gradation of type E6. For (E6, {α4}), we have
Φ+3 = {γ38 = 1 2 3 2 12 , γ37 = 1 2 3 2 11 },
Φ+2 = {γ36 = 1 2 2 2 11 , γ35 = 1 2 2 1 11 , γ34 = 1 1 2 2 11 , γ33 = 1 1 2 1 11 , γ32 = 1 2 2 1 01 ,
γ31 = 0 1 2 2 11 , γ29 =
1 1 2 1 0
1 , γ28 =
0 1 2 1 1
1 , γ24 =
0 1 2 1 0
1 },
Φ+1 = {γ10 = 0 0 1 0 01 , γ14 = 0 0 1 1 01 , γ15 = 0 1 1 0 01 , γ19 = 0 1 1 1 01 , γ20 = 0 0 1 1 11 ,
γ21 = 1 1 1 0 01 , γ25 =
0 1 1 1 1
1 , γ26 =
1 1 1 1 0
1 , γ30 =
1 1 1 1 1
1
γ4 = 0 0 1 0 00 , γ11 =
0 0 1 1 0
0 , γ12 =
0 1 1 0 0
0 , γ16 =
0 1 1 1 0
0 , γ17 =
0 0 1 1 1
0 ,
γ18 = 1 1 1 0 00 , γ22 =
0 1 1 1 1
0 , γ23 =
1 1 1 1 0
0 , γ27 =
1 1 1 1 1
0 }
where a1 a3 a4 a5 a6a2 stands for the root γ =
∑6
i=1 aiαi ∈ Φ+. Also we put γ9 = 0 0 0 1 10 and
γ13 = 1 1 0 0 00 for later use.
We fix the orientation (or sign) of yγ as in the following: First we choose the orientation of
yαi for simple roots by fixing the root vectors yi = yαi ∈ g−αi for i = 1, . . . , 8. For γi ∈ Φ+(i =
13
1, . . . , 38), we put yi = yγi and fix the orientation by the following order;
y9 = [y5, y6], y10 = [y2, y4], y11 = [y4, y5], y12 = [y3, y4], y13 = [y1, y3], y14 = [y2, y11],
y15 = [y2, y12], y16 = [y3, y11], y17 = [y4, y9], y18 = [y4, y13], y19 = [y2, y16], y20 = [y2, y17],
y21 = [y2, y18], y22 = [y3, y17], y23 = [y1, y16], y24 = [y4, y19], y25 = [y2, y22], y26 = [y2, y23],
y27 = [y1, y22], y28 = [y4, y25], y29 = [y2, y26], y30 = [y2, y27], y31 = [y5, y28], y32 = [y3, y29],
y33 = [y4, y30], y34 = [y1, y31], y35 = [y3, y33], y36 = [y3, y34], y37 = [y4, y36], y38 = [y2, y37],
Then, by the repeated application of Jacobi identities, we obtain
y38 = [−y36, y10] = [−y35, y14] = [y34, y15] = [y33, y19] = [y32, y20]
= [y31, y21] = [−y29, y25] = [−y28, y26] = [y24, y30],
y37 = [−y36, y4] = [−y35, y11] = [y34, y12] = [y33, y16] = [y32, y17]
= [y31, y18] = [−y29, y22] = [−y28, y23] = [y24, y27],
Thus, putting
W1 = y38, W2 = y37,
Z1 = −y36, Z2 = −y35, Z3 = y34, Z4 = y33, Z5 = y32,
Z6 = y31, Z7 = −y29, Z8 = −y28, Z9 = y24,
Y1 = y10, Y2 = y14, Y3 = y15, Y4 = y19, Y5 = y20,
Y6 = y21, Y7 = y25, Y8 = y26, Y9 = y30,
X1 = y4, X2 = y11, X3 = y12, X4 = y16, X5 = y17,
X6 = y18, X7 = y22, X8 = y23, X9 = y27,
we obtain the basis {W1,W2, Z1, . . . , Z9, Y1, . . . , Y9, X1, . . . , X9} of m satisfying the following:
g−3 = ⟨{W1,W2}⟩, g−2 = ⟨{Z1, . . . , Z9}⟩, g−1 = ⟨{Y1, . . . , Y9, X1, . . . , X9}⟩
such that
[Zi, Yj] = δ
i
jW1, [Zi, Xj] = δ
i
jW2 [Xi, Xj] = [Yi, Yj] = 0 (1 ≦ i, j ≦ 9)
Then we calculate [Xj, Yk] for 1 ≦ j, k ≦ 9 and obtain
Z1 = [X4, Y9] = −[X7, Y8] = −[X8, Y7] = [X9, Y4],
Z2 = −[X3, Y9] = −[X6, Y7] = −[X7, Y6] = −[X9, Y3],
Z3 = −[X2, Y9] = [X5, Y8] = [X8, Y5] = −[X9, Y2],
Z4 = [X1, Y9] = [X5, Y6] = [X6, Y5] = [X9, Y1],
Z5 = [X3, Y8] = [X4, Y6] = [X6, Y4] = [X8, Y3],
Z6 = −[X2, Y7] = [X4, Y5] = [X5, Y4] = −[X7, Y2],
Z7 = −[X1, Y8] = −[X2, Y6] = −[X6, Y2] = −[X8, Y1],
Z8 = −[X1, Y7] = [X3, Y5] = [X5, Y3] = −[X7, Y1],
Z9 = [X1, Y4] = −[X2, Y3] = −[X3, Y2] = [X4, Y1].
14
Here we define the bilinear forms fi(x1, . . . , x9, y1, . . . , y9)(i = 1, . . . , 9) as follows;
f1 = x4 y9 − x7 y8 − x8 y7 + x9 y4,
f2 = −x3 y9 − x6 y7 − x7 y6 − x9 y3,
f3 = −x2 y9 + x5 y8 + x8 y5 − x9 y2,
f4 = x1 y9 + x5 y6 + x6 y5 + x9 y1,
f5 = x3 y8 + x4 y6 + x6 y4 + x8 y3,
f6 = −x2 y7 + x4 y5 + x5 y4 − x7 y2,
f7 = −x1 y8 − x2 y6 − x6 y2 − x8 y1,
f8 = −x1 y7 + x3 y5 + x5 y3 − x7 y1,
f9 = x1 y4 − x2 y3 − x3 y2 + x4 y1
Moreover we put
9∑
i=1
xi fi = 2
9∑
i=1
yi gi
where the quadratic forms gi(x1, . . . , x9)(i = 1, . . . , 9) are given by
g1 = x4 x9 − x7 x8, g2 = −x3 x9 − x6 x7, g3 = x5 x8 − x2 x9,
g4 = x1 x9 + x5 x6, g5 = x3 x8 + x4 x6, g6 = x4 x5 − x2 x7,
g7 = −x1 x8 − x2 x6, g8 = x3 x5 − x1 x7, g9 = x1 x4 − x2 x3.
Then we have
9∑
i=1
fi dxi =
9∑
i=1
yi dgi,
9∑
i=1
gi dxi = dg, 3g =
9∑
i=1
xi gi,
where g(x1, . . . , x9) is the cubic form given by
g = x1 x4 x9 − x1 x7 x8 − x2 x3 x9 − x2 x6 x7 + x3 x5 x8 + x4 x5 x6.
Thus, by (4.1), we obtain
ω1 = dP2 + (a4a9 − a7a8)dX1 − a9dX5 + a8dX8 + a7dX9 − a4dX10,
ω2 = dP3 − (a3a9 + a6a7)dX1 + a9dX4 + a7dX7 + a6dX8 + a3dX10,
ω3 = dP4 + (a5a8 − a2a9)dX1 + a9dX3 − a8dX6 − a5dX9 + a2dX10,
ω4 = dP5 + (a1a9 + a5a6)dX1 − a9dX2 − a6dX6 − a5dX7 − a1X10,
ω5 = dP6 + (a3a8 + a4a6)dX1 − a8dX4 − a6dX5 − a4dX7 − a3dX9,
ω6 = dP7 + (a4a5 − a2a7)dX1 + a7dX3 − a5dX5 − a4dX6 + a2X8,
ω7 = dP8 − (a1a8 + a2a6)dX1 + a8dX2 + a6dX3 + a2dX7 + a1dX9,
ω8 = dP9 + (a3a5 − a1a7)dX1 + a7dX2 − a5dX4 − a3dX6 + a1dX8,
ω9 = dP10 + (a1a4 − a2a3)dX1 − a4dX2 + a3dX3 + a2dX4 − a1dX5,
ϖ2 = a1ω1 + a2ω2 + · · ·+ a9ω9+
dP1 − 2(a1a4a9 − a1a7a8 − a2a3a9 − a2a6a7 + a3a5a8 + a4a5a6)dX1
+ (a4a9 − a7a8)dX2 − (a3a9 + a6a7)dX3 + (a5a8 − a2a9)dX4
+ (a1a9 + a5a6)dX5 + (a3a8 + a4a6)dX6 + (a4a5 − a2a7)dX7
− (a1a8 + a2a6)dX8 + (a3a5 − a1a7)dX9 + (a1a4 − a2a3)dX10
15
This implies R(X) is given by the following 46 equations;
P5,10 = −P89 (= a1), P78 = P4,10 (= −a2), P69 = −P3,10 (= a3),
P67 = P2,10 (= a4), P49 = P57 (= a5), P56 = −P38 (= a6),
P29 = P37 (= −a7), P46 = −P28 (= a8), P25 = −P34 (= a9)
P22 = P23 = P24 = P26 = P27 = P33 = P35 = P36 = P39 = P44 = P45 = P47 = P48 = P55 = 0,
P58 = P59 = P66 = P68 = P6,10 = P77 = P79 = P7,10 = P88 = P8,10 = P99 = P9,10 = P10,10 = 0,
P1,10 = −P78P69 − P5,10P67, P19 = −P5,10P29 − P69P49, P18 = P5,10P46 − P78P56,
P17 = −P78P29 − P67P49, P16 = −P69P46 − P67P56, P15 = −P5,10P25 − P49P56,
P14 = −P78P25 − P49P46, P13 = P69P25 − P56P29, P12 = −P29P46 − P67P25,
P11 = 2(P5,10P67P25 + P5,10P29P46 + P78P69P25 − P78P56P29 + P69P49P46 + P67P49P56).
in coordinates (X1, . . . , X10, Z, P1, . . . , P10, P11, . . . , P10,10) of L(J).
4.3. Goursat gradation and model system of type E7. Let m be the Goursat gradation
of type E7. For (E7, {α3}), we have
Φ+3 = {γ64 = 2 3 4 3 2 12 , γ63 = 1 3 4 3 2 12 },
Φ+2 = {γ62 = 1 2 4 3 2 12 , γ61 = 1 2 3 3 2 12 , γ60 = 1 2 3 3 2 11 , γ59 = 1 2 3 2 2 12 , γ58 = 1 2 3 2 2 11 ,
γ57 = 1 2 3 2 1 12 , γ56 =
1 2 2 2 2 1
1 , γ55 =
1 2 3 2 1 1
1 , γ53 =
1 2 2 2 1 1
1 , γ50 =
1 2 2 1 1 1
1 ,
γ38 = 1 2 3 2 1 02 , γ37 =
1 2 3 2 1 0
1 , γ36 =
1 2 2 2 1 0
1 , γ35 =
1 2 2 1 1 0
1 , γ32 =
1 2 2 1 0 0
1 , },
Φ+1 = {γ13 = 1 1 0 0 0 00 , γ18 = 1 1 1 0 0 00 , γ21 = 1 1 1 0 0 01 , γ23 = 1 1 1 1 0 00 , γ26 = 1 1 1 1 0 01 ,
γ27 = 1 1 1 1 1 00 , γ29 =
1 1 2 1 0 0
1 , γ30 =
1 1 1 1 1 0
1 , γ33 =
1 1 2 1 1 0
1 , γ34 =
1 1 2 2 1 0
1 ,
γ45 = 1 1 1 1 1 10 , γ47 =
1 1 1 1 1 1
1 , γ48 =
1 1 2 1 1 1
1 , γ51 =
1 1 2 2 1 1
1 , γ54 =
1 1 2 2 2 1
1 ,
γ3 = 0 1 0 0 0 00 , γ12 =
0 1 1 0 0 0
0 , γ15 =
0 1 1 0 0 0
1 , γ16 =
0 1 1 1 0 0
0 , γ19 =
0 1 1 1 0 0
1 ,
γ22 = 0 1 1 1 1 00 , γ24 =
0 1 2 1 0 0
1 , γ25 =
0 1 1 1 1 0
1 , γ28 =
0 1 2 1 1 0
1 , γ31 =
0 1 2 2 1 0
1 ,
γ43 = 0 1 1 1 1 10 , γ44 =
0 1 1 1 1 1
1 , γ46 =
0 1 2 1 1 1
1 , γ49 =
0 1 2 2 1 1
1 , γ52 =
0 1 2 2 2 1
1 , },
where a1 a3 a4 a5 a6 a7a2 stands for the root γ =
∑7
i=1 aiαi ∈ Φ+.
We fix the orientation (or sign) of yγ as in the following: For γi ∈ Φ+(i = 39, . . . , 64), we put
yi = yγi and fix the orientation by the following order;
y39 = [y6, y11], y40 = [y5, y39], y41 = [y4, y40], y42 = [y2, y41], y43 = [y7, y22], y44 = [y2, y43],
y45 = [y7, y27], y46 = [y4, y44], y47 = [y2, y45], y48 = [y4, y47], y49 = [y7, y31], y50 = [y7, y35],
y51 = [y7, y34], y52 = [y6, y49], y53 = [y5, y50], y54 = [y6, y51], y55 = [y4, y53], y56 = [y6, y53],
y57 = [y2, y55], y58 = [y4, y56], y59 = [y2, y58], y60 = [y5, y58], y61 = [y2, y60], y62 = [y4, y61],
y63 = [y3, y62], y64 = [y1, y63],
Then, by the repeated application of Jacobi identities, we obtain
y64 = [−y62, y13] = [y61, y18] = [y60,−y21] = [−y59, y23] = [y58, y26] = [−y57, y27] = [−y56, y29]
= [y55, y30] = [−y53, y33] = [y50, y34] = [y38, y45] = [−y37, y47] = [y36, y48] = [−y35, y51] = [−y32, y54]
y63 = [−y62, y3] = [y61,−y12] = [y60, y15] = [−y59, y16] = [y58, y19] = [−y57, y22] = [−y56, y24]
= [y55, y25] = [−y53, y28] = [y50, y31] = [y38, y43] = [−y37, y44] = [y36, y46] = [−y35, y49] = [−y32, y52]
16
Thus, putting
W1 = y64, W2 = y63,
Z1 = −y62, Z2 = y61, Z3 = y60, Z4 = −y59, Z5 = y58, Z6 = −y57, Z7 = −y56, Z8 = y55,
Z9 = −y53, Z10 = y50, Z11 = y38, Z12 = −y37, Z13 = y36, Z14 = −y35, Z15 = −y32,
Y1 = y13, Y2 = y18, Y3 = −y21, Y4 = y23, Y5 = y26, Y6 = y27, Y7 = y29, Y8 = y30,
Y9 = y33, Y10 = y34, Y11 = y45, Y12 = y47, Y13 = y48, Y14 = y51, Y15 = y54,
X1 = y3, X2 = −y12, X3 = y15, X4 = y16, X5 = y19, X6 = y22, X7 = y24, X8 = y25,
X9 = y28, X10 = y31, X11 = y43, X12 = y44, X13 = y46, X14 = y49, X15 = y52.
we obtain the basis {W1,W2, Z1, . . . , Z15, Y1, . . . , Y15, X1, . . . , X15} of m satisfying the following:
g−3 = ⟨{W1,W2}⟩, g−2 = ⟨{Z1, . . . , Z15}⟩, g−1 = ⟨{Y1, . . . , Y15, X1, . . . , X15}⟩
such that
[Zi, Yj] = δ
i
jW1, [Zi, Xj] = δ
i
jW2 [Xi, Xj] = [Yi, Yj] = 0 (1 ≦ i, j ≦ 15)
Then we calculate [Xj, Yk] for 1 ≦ j, k ≦ 15 and obtain
Z1 = −[X7, Y15] = −[X9, Y14] = [X10, Y13] = [X13, Y10] = −[X14, Y9] = −[X15, Y7]
Z2 = [X5, Y15] = [X8, Y14] = −[X10, Y12] = −[X12, Y10] = [X14, Y8] = [X15, Y5],
Z3 = [X4, Y15] = [X6, Y14] = −[X10, Y11] = −[X11, Y10] = [X14, Y6] = [X15, Y4],
Z4 = [X3, Y15] = −[X8, Y13] = [X9, Y12] = [X12, Y9] = −[X13, Y8] = [X15, Y3],
Z5 = [X2, Y15] = [X6, Y13] = −[X9, Y11] = −[X11, Y9] = [X13, Y6] = [X15, Y2],
Z6 = [X3, Y14] = [X5, Y13] = −[X7, Y12] = −[X12, Y7] = [X13, Y5] = [X14, Y3],
Z7 = −[X1, Y15] = −[X6, Y12] = [X8, Y11] = [X11, Y8] = −[X12, Y6] = −[X15, Y1]
Z8 = [X2, Y14] = −[X4, Y13] = [X7, Y11] = [X11, Y7] = −[X13, Y4] = [X14, Y2],
Z9 = −[X1, Y14] = [X4, Y12] = −[X5, Y11] = −[X11, Y5] = [X12, Y4] = −[X14Y1]
Z10 = [X1, Y13] = −[X2, Y12] = −[X3, Y11] = −[X11, Y3] = −[X12, Y2] = [X13, Y1]
Z11 = −[X3, Y10] = −[X5, Y9] = [X7, Y8] = [X8, Y7] = −[X9, Y5] = −[X10, Y3],
Z12 = −[X2, Y10] = [X4, Y9] = −[X6, Y7] = −[X7, Y6] = [X9, Y4] = −[X10, Y2]
Z13 = [X1, Y10] = −[X4, Y8] = [X5, Y6] = [X6, Y5] = −[X8, Y4] = [X10, Y1]
Z14 = −[X1, Y9] = [X2, Y8] = [X3, Y6] = [X6, Y3] = [X8, Y2] = −[X9, Y1]
Z15 = −[X1, Y7] = [X2, Y5] = [X3, Y4] = [X4, Y3] = [X5, Y2] = −[X7, Y1],
17
Here we define the bilinear forms fi(x1, . . . , x15, y1, . . . , y15)(i = 1, . . . , 15) as follows;

f1 = −x7 y15 − x9 y14 + x10 y13 + x13 y10 − x14 y9 − x15 y7,
f2 = x5 y15 + x8 y14 − x10 y12 − x12 y10 + x14 y8 + x15 y5,
f3 = x4 y15 + x6 y14 − x10 y11 − x11 y10 + x14 y6 + x15 y4,
f4 = x3 y15 − x8 y13 + x9 y12 + x12 y9 − x13 y8 + x15 y3,
f5 = x2 y15 + x6 y13 − x9 y11 − x11 y9 + x13 y6 + x15 y2,
f6 = x3 y14 + x5 y13 − x7 y12 − x12 y7 + x13 y5 + x14 y3,
f7 = −x1 y15 − x6 y12 + x8 y11 + x11 y8 − x12 y6 − x15 y1
f8 = x2 y14 − x4 y13 + x7 y11 + x11 y7 − x13 y4 + x14 y2,
f9 = −x1 y14 + x4 y12 − x5 y11 − x11 y5 + x12 y4 − x14 y1
f10 = x1 y13 − x2 y12 − x3 y11 − x11 y3 − x12 y2 + x13 y1
f11 = −x3 y10 − x5 y9 + x7 y8 + x8 y7 − x9 y5 − x10 y3,
f12 = −x2 y10 + x4 y9 − x6 y7 − x7 y6 + x9 y4 − x10 y2
f13 = x1 y10 − x4 y8 + x5 y6 + x6 y5 − x8 y4 + x10 y1
f14 = −x1 y9 + x2 y8 + x3 y6 + x6 y3 + x8 y2 − x9y1
f15 = −x1 y7 + x2 y5 + x3 y4 + x4 y3 + x5 y2 − x7 y1
Moreover we put
15∑
i=1
xi fi = 2
15∑
i=1
yi gi
where the quadratic forms gi(x1, . . . , x15)(i = 1, . . . , 15) are given by

g1 = −x7 x15 − x9 x14 + x10 x13, g2 = x5 x15 + x8 x14 − x10 x12, g3 = x4 x15 + x6 x14 − x10 x11,
g4 = x3 x15 − x8 x13 + x9 x12, g5 = x2 x15 + x6 x13 − x9 x11, g6 = x3 x14 + x5 x13 − x7 x12,
g7 = −x1 x15 − x6 x12 + x8 x11, g8 = x2 x14 − x4 x13 + x7 x11, g9 = −x1 x14 + x4 x12 − x5 x11,
g10 = x1 x13 − x2 x12 − x3 x11, g11 = −x3 x10 − x5 x9 + x7 x8, g12 = −x2 x10 + x4 x9 − x6 x7,
g13 = x1 x10 − x4 x8 + x5 x6, g14 = −x1 x9 + x2 x8 + x3 x6, g15 = −x1 x7 + x2 x5 + x3 x4.
Then we have
15∑
i=1
fi dxi =
15∑
i=1
yi dgi,
15∑
i=1
gi dxi = dg, 3g =
15∑
i=1
xi gi,
where g(x1, . . . , x15) is the cubic form given by
g = −x1 x7 x15 − x1 x9 x14 + x1 x10 x13 + x2 x5 x15 + x2 x8 x14 − x2 x10 x12. + x3 x4 x15
+ x3 x6 x14 − x3 x10 x11 − x4 x8 x13 + x4 x9 x12 + x5 x6 x13 − x5 x9 x11 − x6 x7 x12 + x7 x8 x11.
18
Thus, by (4.1), we obtain
ω1 = dP2 + ĝ1dX1 − (−a15dX8 − a14dX10 + a13dX11 + a10dX14 − a9dX15 − a7dX16),
ω2 = dP3 + ĝ2dX1 − (a15dX6 + a14dX9 − a12dX11 − a10dX13 + a8dX15 + a5dX16),
ω3 = dP4 + ĝ3dX1 − (a15dX5 + a14dX7 − a11dX11 − a10dX12 + a6dX15 + a4dX16),
ω4 = dP5 + ĝ4dX1 − (a15dX4 − a13dX9 + a12dX10 + a9dX13 − a8dX14 + a3dX16),
ω5 = dP6 + ĝ5dX1 − (a15dX3 + a13dX7 − a11dX10 − a9X12 + a6dX14 + a2dX16),
ω6 = dP7 + ĝ6dX1 − (a14dX4 + a13dX6 − a12dX8 − a7dX13 + a5dX14 + a3dX15),
ω7 = dP8 + ĝ7dX1 − (−a15dX2 − a12dX7 + a11dX9 + a8dX12 − a6dX13 − a1dX16),
ω8 = dP9 + ĝ8dX1 − (a14dX3 − a13dX5 + a11dX8 + a7dX12 − a4dX14 + a2dX15),
ω9 = dP10 + ĝ9dX1 − (−a14dX2 + a12dX5 − a11dX6 − a5dX12 + a4dX13 − a1dX15),
ω10 = dP11 + ĝ10dX1 − (a13dX2 − a12dX3 − a11dX4 − a3dX12 − a2dX13 + a1dX14),
ω11 = dP12 + ĝ11dX1 − (−a10dX4 − a9dX6 + a8dX8 + a7dX9 − a5dX10 − a3dX11),
ω12 = dP13 + ĝ12dX1 − (−a10dX3 + a9dX5 − a7dX7 − a6dX8 + a4dX10 − a2dX11),
ω13 = dP14 + ĝ13dX1 − (a10dX2 − a8dX5 + a6dX6 + a5dX7 − a4dX9 + a1dX11),
ω14 = dP15 + ĝ14dX1 − (−a9dX2 + a8dX3 + a6dX4 + a3dX7 + a2dX9 − a1dX10),
ω15 = dP16 + ĝ15dX1 − (−a7dX2 + a5dX3 + a4dX4 + a3dX5 + a2dX6 − a1dX8),
ϖ2 = a1ω1 + a2ω2 + · · ·+ a15ω15 + dP1 − 2ĝdX1 + ĝ1dX2.+ · · ·+ ĝ15dX16
This implies R(X) is given by the following 121 equations;
−P8,16 = −P10,15 = P11,14 (= a1), P6,16 = P9,15 = −P11,13 (= a2), P5,16 = P7,15 = −P11,12 (= a3),
P4,16 = −P9,14 = P10,13 (= a4), P3,16 = P7,14 = −P10,12 (= a5), P4,15 = P6,14 = −P8,13 (= a6),
−P2,16 = −P7,13 = P9,12 (= a7), P3,15 = −P5,14 = P8,12 (= a8), −P2,15 = P5,13 = −P6,12 (= a9),
P2,14 = −P3,13 = −P4,12 (= a10), −P4,11 = −P6,10 = P8,9 (= a11), −P3,11 = P5,10 = −P7,8 (= a12),
P2,11 = −P5,9 = P6,7 (= a13), −P2,10 = P3,9 = P4,7 (= a14), −P2,8 = P3,6 = P4,5 (= a15),
P2,2 = P2,3 = P2,4 = P2,5 = P2,6 = P2,7 = P2,9 = P2,12 = P2,13 = P3,3 = P3,4 = P3,5 = P3,7 = 0,
P3,8 = P3,10 = P3,12 = P3,14 = P4,4 = P4,6 = P4,8 = P4,9 = P4,10 = P4,13 = P4,14 = P5,5 = 0,
P5,6 = P5,7 = P5,8 = P5,11 = P5,12 = P5,15 = P6,6 = P6,8 = P6,9 = P6,11 = P6,13 = P6,15 = 0,
P7,7 = P7,9 = P7,10 = P7,11 = P7,12 = P7,16 = P8,8 = P8,10 = P8,11 = P8,14 = P8,15 = 0,
P9,9 = P9,10 = P9,11 = P9,13 = P9,16 = P10,10 = P10,11 = P10,14 = P10,16 = 0,
P11,11 = P11,15 = P11,16 = P12,12 = P12,13 = P12,14 = P12,15 = P12,16 = 0,
P13,13 = P13,14 = P13,15 = P13,16 = P14,14 = P14,15 = P14,16 = P15,15 = P15,16 = P16,16 = 0
P1,16 = P2,16P8,16 − P3,16P6,16 − P4,16P5,16, P1,15 = −P2,15P8,16 − P3,15P6,16 − P4,15P5,16,
P1,14 = P2,14P8,16 + P3,15P4,16 − P3,16P4,15, P1,13 = P2,14P5,16 + P2,15P4,16 − P2,16P4,15,
P1,12 = P2,14P5,16 − P2,15P3,16 + P2,15P2,16, P1,11 = P2,11P8,16 − P3,11P6,16 + P4,11P5,16,
P1,10 = P2,10P8,16 + P3,11P4,16 − P3,16P4,11, P1,9 = P2,10P6,16 + P2,11P4,16 − P2,16P4,11,
P1,8 = P2,8P8,16 − P2,11P4,15 + P3,15P8,9, P1,7 = P2,10P5,16 − P2,11P3,16 + P2,16P3,11,
P1,6 = P2,8P6,16 − P2,11P4,15 + P2,16P4,11, P1,5 = P2,8P5,16 + P2,11P3,15 − P2,15P3,11,
P1,4 = P2,8P4,16 + P2,10P4,15 − P2,14P4,11, P1,3 = P2,8P3,16 + P2,10P3,15 − P2,14P3,11,
P1,2 = P2,8P2,16 + P2,10P2,15 − P2,11P2,14
19
P1,1 = 2(P2,8P2,16P8,16 + P2,10P2,15P8,16 − P2,11P2,14P8,16 − P2,8P3,16P6,16 − P2,10P3,15P6,16
+ P2,14P3,11P6,16 − P2,8P4,16P5,16 − P2,10P4,15P5,16 + P2,14P4,11P5,16 − P2,11P3,15P4,16
+ P2,15P3,11P4,16 + P2,11P3,16P4,15 − P2,15P3,16P4,11 − P2,16P3,11P4,15 + P2,16P3,15P4,11),
in coordinates (X1, . . . , X16, Z, P1, . . . , P16, P11, . . . , P16,16) of L(J).
4.4. Goursat gradation and model system of type E8. Let m be the Goursat gradation
of type E8. For (E8, {α7}), we have
Φ+3 = {γ120 = 2 4 6 5 4 3 23 , γ119 = 2 4 6 5 4 3 13 },
Φ+2 = {γ118 = 2 4 6 5 4 2 13 , γ117 = 2 4 6 5 3 2 13 , γ116 = 2 4 6 4 3 2 13 , γ115 = 2 4 5 4 3 2 13 , γ114 = 2 4 5 4 3 2 12 ,
γ113 = 2 3 5 4 3 2 13 , γ112 =
2 3 5 4 3 2 1
2 , γ111 =
1 3 5 4 3 2 1
3 , γ110 =
2 3 4 4 3 2 1
2 , γ109 =
1 3 5 4 3 2 1
2 ,
γ108 = 1 3 4 4 3 2 12 , γ107 =
2 3 4 3 3 2 1
2 , γ106 =
1 2 4 4 3 2 1
2 , γ105 =
1 3 4 3 3 2 1
2 , γ104 =
2 3 4 3 2 2 1
2 ,
γ103 = 1 2 4 3 3 2 12 , γ102 =
1 3 4 3 2 2 1
2 , γ100 =
1 2 3 3 3 2 1
2 , γ99 =
1 2 4 3 2 2 1
2 , γ97 =
1 2 3 3 3 2 1
1 ,
γ96 = 1 2 3 3 2 2 12 , γ94 =
1 2 3 3 2 2 1
1 , γ93 =
1 2 3 2 2 2 1
2 , γ91 =
1 2 3 2 2 2 1
1 , γ88 =
1 2 2 2 2 2 1
1 ,
γ85 = 1 1 2 2 2 2 11 , γ82 =
0 1 2 2 2 2 1
1 , },
Φ+1 = {γ65 = 0 0 0 0 0 1 10 , γ66 = 0 0 0 0 1 1 10 , γ67 = 0 0 0 1 1 1 10 , γ68 = 0 0 1 1 1 1 10 , γ69 = 0 0 1 1 1 1 11 ,
γ70 = 0 1 1 1 1 1 10 , γ71 =
0 1 1 1 1 1 1
1 , γ72 =
1 1 1 1 1 1 1
0 , γ73 =
0 1 2 1 1 1 1
1 , γ74 =
1 1 1 1 1 1 1
1 ,
γ75 = 1 1 2 1 1 1 11 , γ76 =
0 1 2 2 1 1 1
1 , γ77 =
1 2 2 1 1 1 1
1 , γ78 =
1 1 2 2 1 1 1
1 , γ79 =
0 1 2 2 2 1 1
1 ,
γ80 = 1 2 2 2 1 1 11 , γ81 =
1 1 2 2 2 1 1
1 , γ83 =
1 2 3 2 1 1 1
1 , γ84 =
1 2 2 2 2 1 1
1 , γ86 =
1 2 3 2 1 1 1
2 ,
γ87 = 1 2 3 2 2 1 11 , γ89 =
1 2 3 2 2 1 1
2 , γ90 =
1 2 3 3 2 1 1
1 , γ92 =
1 2 3 3 2 1 1
2 , γ95 =
1 2 4 3 2 1 1
2 ,
γ98 = 1 3 4 3 2 1 12 , γ101 =
2 3 4 3 2 1 1
2 ,
γ7 = 0 0 0 0 0 1 00 , γ39 =
0 0 0 0 1 1 0
0 , γ40 =
0 0 0 1 1 1 0
0 , γ41 =
0 0 1 1 1 1 0
0 , γ42 =
0 0 1 1 1 1 0
1 ,
γ43 = 0 1 1 1 1 1 00 , γ44 =
0 1 1 1 1 1 0
1 , γ45 =
1 1 1 1 1 1 0
0 , γ46 =
0 1 2 1 1 1 0
1 , γ47 =
1 1 1 1 1 1 0
1 ,
γ48 = 1 1 2 1 1 1 01 , γ49 =
0 1 2 2 1 1 0
1 , γ50 =
1 2 2 1 1 1 0
1 , γ51 =
1 1 2 2 1 1 0
1 , γ52 =
0 1 2 2 2 1 0
1 ,
γ53 = 1 2 2 2 1 1 01 , γ54 =
1 1 2 2 2 1 0
1 , γ55 =
1 2 3 2 1 1 0
1 , γ56 =
1 2 2 2 2 1 0
1 , γ57 =
1 2 3 2 1 1 0
2 ,
γ58 = 1 2 3 2 2 1 01 , γ59 =
1 2 3 2 2 1 0
2 , γ60 =
1 2 3 3 2 1 0
1 , γ61 =
1 2 3 3 2 1 0
2 , γ62 =
1 2 4 3 2 1 0
2 ,
γ63 = 1 3 4 3 2 1 02 , γ64 =
2 3 4 3 2 1 0
2 },
where a1 a3 a4 a5 a6 a7 a8a2 stands for the root γ =
∑8
i=1 aiαi ∈ Φ+.
We fix the orientation (or sign) of yγ as in the following: For γi ∈ Φ+(i = 65, . . . , 120), we
put yi = yγi and fix the orientation by the following order;
y65 = [y7, y8], y66 = [y6, y65], y67 = [y5, y66], y68 = [y4, y67], y69 = [y2, y68], y70 = [y3, y68],
y71 = [y2, y70], y72 = [y1, y70], y73 = [y4, y71], y74 = [y1, y71], y75 = [y1, y73], y76 = [y5, y73],
y77 = [y3, y75], y78 = [y5, y75], y79 = [y6, y76], y80 = [y5, y77], y81 = [y6, y78], y82 = [y7, y79],
y83 = [y7, y80], y84 = [y6, y80], y85 = [y1, y82], y86 = [y2, y83], y87 = [y6, y83], y88 = [y3, y85],
y89 = [y2, y87], y90 = [y5, y87], y91 = [y4, y88], y92 = [y7, y90], y93 = [y7, y91], y94 = [y5, y91],
y95 = [y4, y92], y96 = [y2, y94], y97 = [y6, y94], y98 = [y3, y95], y99 = [y4, y96], y100 = [y2, y97],
y101 = [y1, y98], y102 = [y3, y99], y103 = [y4, y100], y104 = [y1, y102], y105 = [y3, y103],
y106 = [y5, y103], y107 = [y1, y105], y108 = [y5, y105], y109 = [y4, y108], y110 = [y1, y108],
20
y111 = [y2, y109], y112 = [y4, y110], y113 = [y2, y112], y114 = [y3, y112], y115 = [y2, y114],
y116 = [y4, y115], y117 = [y5, y116], y118 = [y6, y117], y119 = [y7, y118], y120 = [y8, y119].
Then, by the repeated application of Jacobi identities, we obtain
y120 = [−y118, y65] = [y117, y66] = [−y116, y67] = [y115, y68] = [y114,−y69] = [−y113, y70] = [y112, y71]
= [y111, y72] = [y110,−y73] = [y109,−y74] = [y108, y75] = [y107, y76] = [y106,−y77] = [y105,−y78]
= [y104,−y79] = [y103, y80] = [y102, y81] = [y100,−y83] = [y99,−y84] = [y97, y86] = [y96, y87]
= [y94,−y89] = [y93,−y90] = [y91, y92] = [y88,−y95] = [y85, y98] = [y82,−y101]
y119 = [−y118, y7] = [y117, y39] = [−y116, y40] = [y115, y41] = [y114, y42] = [−y113, y43] = [y112,−y44]
= [y111,−y45] = [y110, y46] = [y109, y47] = [y108,−y48] = [y107,−y49] = [y106, y50] = [y105, y51]
= [y104, y52] = [y103,−y53] = [y102,−y54] = [y100, y55] = [y99, y56] = [y97,−y57] = [y96,−y58]
= [y94, y59] = [y93, y60] = [y91,−y61] = [y88, y62] = [y85,−y63] = [y82, y64]
Thus, putting
W1 = y120, W2 = y119,
Z1 = −y118, Z2 = y117, Z3 = −y116, Z4 = y115, Z5 = y114, Z6 = −y113, Z7 = y112,
Z8 = y111, Z9 = y110, Z10 = y109, Z11 = y108, Z12 = y107, Z13 = y106, Z14 = y105,
Z15 = y104, Z16 = y103, Z17 = y102, Z18 = y100, Z19 = y99, Z20 = y97, Z21 = y96,
Z22 = y94, Z23 = y93, Z24 = y91, Z25 = y88, Z26 = y85, Z27 = y82,
Y1 = y65, Y2 = y66, Y3 = y67, Y4 = y68, Y5 = −y69, Y6 = y70, Y7 = y71,
Y8 = y72, Y9 = −y73, Y10 = −y74, Y11 = y75, Y12 = y76, Y13 = −y77, Y14 = −y78,
Y15 = −y79, Y16 = y80, Y17 = y81, Y18 = −y83, Y19 = −y84, Y20 = y86, Y21 = y87,
Y22 = −y89, Y23 = −y90, Y24 = y92, Y25 = −y95, Y26 = y98, Y27 = −y101,
X1 = y7, X2 = y39, X3 = y40, X4 = y41, X5 = y42, X6 = y43, X7 = −y44,
X8 = −y45, X9 = y46, X10 = y47, X11 = −y48, X12 = −y49, X13 = y50, X14 = y51,
X15 = y52, X16 = −y53, X17 = −y54, X18 = y55, X19 = y56, X20 = −y57, X21 = −y58,
X22 = y59, X23 = y60, X24 = −y61, X25 = y62, X26 = −y63, X27 = y64,
we obtain the basis {W1,W2, Z1, . . . , Z27, Y1, . . . , Y27, X1, . . . , X27} of m satisfying the following:
g−3 = ⟨{W1,W2}⟩, g−2 = ⟨{Z1, . . . , Z27}⟩, g−1 = ⟨{Y1, . . . , Y27, X1, . . . , X27}⟩
such that
[Zi, Yj] = δ
i
jW1, [Zi, Xj] = δ
i
jW2 [Xi, Xj] = [Yi, Yj] = 0 (1 ≦ i, j ≦ 27)
21
Then we calculate [Xj, Yk] for 1 ≦ j, k ≦ 27 and obtain
Z1 = −[X13, Y27] = [X16, Y26] = −[X18, Y25] = [X20, Y24] = −[X22, Y23]
= −[X23, Y22] = [X24, Y20] = −[X25, Y18] = [X26, Y16] = −[X27, Y13],
Z2 = −[X11, Y27] = [X14, Y26] = −[X17, Y25] = [X19, Y24] = −[X21, Y22]
= −[X22, Y21] = [X24, Y19] = −[X25, Y17] = [X26, Y14] = −[X27, Y11],
Z3 = −[X9, Y27] = [X12, Y26] = −[X15, Y25] = [X19, Y23] = −[X20, Y21]
= −[X21, Y20] = [X23, Y19] = −[X25, Y15] = [X26, Y12] = −[X27, Y9],
Z4 = −[X7, Y27] = [X10, Y26] = −[X15, Y24] = [X17, Y23]− [X18, Y21]
= −[X21, Y18] = [X23, Y17] = −[X24, Y15] = [X26, Y10] = −[X27, Y7],
Z5 = −[X6, Y27] = −[X10, Y25] = [X12, Y24] = −[X14, Y23] = [X16, Y21]
= [X21, Y16] = −[X23, Y14] = [X24, Y12] = −[X25, Y10] = −[X27, Y6],
Z6 = −[X5, Y27] = [X8, Y26] = −[X15, Y22] = [X17, Y20] = −[X18, Y19]
= −[X19, Y18] = [X20, Y17] = −[X22, Y15] = [X26, Y8] = −[X27, Y5],
Z7 = −[X4, Y27] = [X8, Y25] = −[X12, Y22] = [X14, Y20] = −[X16, Y19]
= −[X19, Y16] = [X20, Y14] = −[X22, Y12] = [X25, Y8] = −[X27, Y4],
Z8 = [X6, Y26] = [X7, Y25] = −[X9, Y24] = [X11, Y23] = −[X13, Y21]
= −[X21, Y13] = [X23, Y11] = −[X24, Y9] = [X25, Y7] = [X26, Y6],
Z9 = −[X3, Y27] = −[X8, Y24] = [X10, Y22] = −[X14, Y18] = [X16, Y17]
= [X17, Y16] = −[X18, Y14] = [X22, Y10] = −[X24, Y8] = −[X27, Y3],
Z10 = [X4, Y26] = −[X5, Y25] = [X9, Y22] = −[X11, Y20] = [X13, Y19]
= [X19, Y13] = −[X20, Y11] = [X22, Y9] = −[X25, Y5] = [X26, Y4],
Z11 = −[X2, Y27] = [X8, Y23] = −[X10, Y20] = [X12, Y18] = −[X15, Y16]
= −[X16, Y15] = [X18, Y12] = −[X20, Y10] = [X23, Y8] = −[X27, Y2],
Z12 = [X3, Y26] = [X5, Y24] = −[X7, Y22] = [X11, Y18] = −[X13, Y17]
= −[X17, Y13] = [X18, Y11] = −[X22, Y7] = [X24, Y5] = [X26, Y3],
Z13 = −[X1, Y27] = −[X8, Y21] = [X10, Y19] = −[X12, Y17] = [X14, Y15]
= [X15, Y14] = −[X17, Y12] = [X19, Y10] = −[X21, Y8] = −[X27, Y1],
Z14 = [X2, Y26] = −[X5, Y23] = [X7, Y20] = −[X9, Y18] = [X13, Y15]
= [X15, Y13] = −[X18, Y9] = [X20, Y7] = −[X23, Y5] = [X26, Y2],
Z15 = −[X3, Y25] = −[X4, Y24] = −[X6, Y22] = −[X11, Y16] = [X13, Y14]
= [X14, Y13] = −[X16, Y11] = −[X22, Y6] = −[X24, Y4] = −[X25, Y3],
Z16 = [X1, Y26] = [X5, Y21] = −[X7, Y19] = [X9, Y17] = −[X11, Y15]
= −[X15, Y11] = [X17, Y9] = −[X19, Y7] = [X21, Y5] = [X26, Y1],
Z17 = −[X2, Y25] = [X4, Y23] = [X6, Y20] = [X9, Y16] = −[X12, Y13]
= −[X13, Y12] = [X16, Y9] = [X20, Y6] = [X23, Y4] = −[X25, Y2],
Z18 = −[X1, Y25] = −[X4, Y21] = −[X6, Y19] = −[X9, Y14] = [X11, Y12]
= [X12, Y11] = −[X14, Y9] = −[X19, Y6] = −[X21, Y4] = −[X25, Y1],
22
Z19 = [X2, Y24] = [X3, Y23] = −[X6, Y18] = −[X7, Y16] = [X10, Y13]
= [X13, Y10] = −[X16, Y7] = −[X18, Y6] = [X23, Y3] = [X24, Y2],
Z20 = [X1, Y24] = −[X3, Y21] = [X6, Y17] = [X7, Y14] = −[X10, Y11]
= −[X11, Y10] = [X14, Y7] = [X17, Y6] = −[X21, Y3] = [X24, Y1],
Z21 = −[X2, Y22] = −[X3, Y20] = −[X4, Y18] = [X5, Y16] = −[X8, Y13]
= −[X13, Y8] = [X16, Y5] = −[X18, Y4] = −[X20Y3] = −[X22, Y2],
Z22 = −[X1, Y23] = −[X2, Y21] = −[X6, Y15] = −[X7, Y12] = [X9, Y10]
= [X10, Y9] = −[X12, Y7] = −[X15, Y6] = −[X21, Y2] = −[X23, Y1],
Z23 = −[X1, Y22] = [X3, Y19] = [X4, Y17] = −[X5, Y14] = [X8, Y11]
= [X11, Y8] = −[X14, Y5] = [X17, Y4] = [X19, Y3] = −[X22, Y1],
Z24 = [X1, Y20] = [X2, Y19] = −[X4, Y15] = [X5, Y12] = −[X8, Y9]
= −[X9, Y8] = [X12, Y5] = −[X15, Y4] = [X19, Y2] = [X20, Y1],
Z25 = −[X1, Y18] = −[X2, Y17] = −[X3, Y15] = −[X5, Y10] = [X7, Y8]
= [X8, Y7] = −[X10, Y5] = −[X15, Y3] = −[X17, Y2] = −[X18, Y1],
Z26 = [X1, Y16] = [X2, Y14] = [X3, Y12] = [X4, Y10] = [X6, Y8]
= [X8, Y6] = [X10, Y4] = [X12, Y3] = [X14, Y2] = [X16, Y1],
Z27 = −[X1, Y13] = −[X2, Y11]−−[X3, Y9] = −[X4, Y7] = −[X5, Y6]
= −[X6, Y5] = −[X7, Y4] = −[X9, Y3] = −[X11, Y2] = −[X13, Y1].
Here we define the bilinear forms fi(x1, . . . , x27, y1, . . . , y27)(i = 1, . . . , 27) as follows;
f1 = −x13 y27 + x16 y26 − x18 y25 + x20 y24 − x22 y23 − x23 y22 + x24 y20 − x25 y18 + x26 y16 − x27 y13,
f2 = −x11 y27 + x14 y26 − x17 y25 + x19 y24 − x21 y22 − x22 y21 + x24 y19 − x25 y17 + x26 y14 − x27 y11,
f3 = −x9 y27 + x12 y26 − x15 y25 + x19 y23 − x20 y21 − x21 y20 + x23 y19 − x25 y15 + x26 y12 − x27 y9,
f4 = −x7 y27 + x10 y26 − x15 y24 + x17 y23 − x18 y21 − x21 y18 + x23 y17 − x24 y15 + x26 y10 − x27 y7,
f5 = −x6 y27 − x10 y25 + x12 y24 − x14 y23 + x16 y21 + x21 y16 − x23 y14 + x24 y12 − x25 y10 − x27 y6,
f6 = −x5 y27 + x8 y26 − x15 y22 + x17 y20 − x18 y19 − x19 y18 + x20 y17 − x22 y15 + x26 y8 − x27 y5,
f7 = −x4 y27 + x8 y25 − x12 y22 + x14 y20 − x16 y19 − x19 y16 + x20 y14 − x22 y12 + x25 y8 − x27 y4,
f8 = x6 y26 + x7 y25 − x9 y24 + x11 y23 − x13 y21 − x21 y13 + x23 y11 − x24 y9 + x25 y7 − x26 y6,
f9 = −x3 y27 − x8 y24 + x10 y22 − x14 y18 + x16 y17 + x17 y16 − x18 y14 + x22 y10 − x24 y8 − x27 y3,
f10 = x4 y26 − x5 y25 + x9 y22 − x11 y20 + x13 y19 + x19 y13 − x20 y11 + x22 y9 − x25 y5 − x26 y4,
f11 = −x2 y27 + x8 y23 − x10 y20 + x12 y18 − x15 y16 − x16 y15 + x18 y12 − x20 y10 + x23 y8 − x27 y2,
f12 = x3 y26 + x5 y24 − x7 y22 + x11 y18 − x13 y17 − x17 y13 + x18 y11 − x22 y7 + x24 y5 + x26 y3,
f13 = −x1 y27 − x8 y21 + x10 y19 − x12 y17 + x14 y15 + x15 y14 − x17 y12 + x19 y10 − x21 y8 − x27 y1,
f14 = x2 y26 − x5 y23 + x7 y20 − x9 y18 + x13 y15 + x15 y13 − x18 y9 + x20 y7 − x23 y5 + x26 y2,
f15 = −x3 y25 − x4 y24 − x6 y22 − x11 y16 + x13 y14 + x14 y13 − x16 y11 − x22 y6 − x24 y4 − x25 y3,
f16 = x1 y26 + x5 y21 − x7 y19 + x9 y17 − x11 y15 − x15 y11 + x17 y9 − x19 y7 + x21 y5 + x26 y1,
f17 = −x2 y25 + x4 y23 + x6 y20 + x9 y16 − x12 y13 − x13 y12 + x16 y9 + x20 y6 + x23 y4 − x25 y2,
f18 = −x1 y25 − x4 y21 − x6 y19 − x9 y14 + x11 y12 + x12 y11 − x14 y9 − x19 y6 − x21 y4 − x25 y1,
23
f19 = x2 y24 + x3 y23 − x6 y18 − x7 y16 + x10 y13 + x13 y10 − x16 y7 − x18 y6 + x23 y3 + x24 y2,
f20 = x1 y24 − x3 y21 + x6 y17 + x7 y14 − x10 y11 − x11 y10 + x14 y7 + x17 y6 − x21 y3 + x24 y1,
f21 = −x2 y22 − x3 y20 − x4 y18 + x5 y16 − x8 y13 − x13 y8 + x16 y5 − x18 y4 − x20 y3 − x22 y2,
f22 = −x1 y23 − x2 y21 − x6 y15 − x7 y12 + x9 y10 + x10 y9 − x12 y7 − x15 y6 − x21 y2 − x23 y1,
f23 = −x1 y22 + x3 y19 + x4 y17 − x5 y14 + x8 y11 + x11 y8 − x14 y5 + x17 y4 + x19 y3 − x22 y1,
f24 = x1 y20 + x2 y19 − x4 y15 + x5 y12 − x8 y9 − x9 y8 + x12 y5 − x15 y4 + x19 y2 + x20 y1,
f25 = −x1 y18 − x2 y17 − x3 y15 − x5 y10 + x7 y8 + x8 y7 − x10 y5 − x15 y3 − x17 y2 − x18 y1,
f26 = x1 y16 + x2 y14 + x3 y12 + x4 y10 + x6 y8 + x8 y6 + x10 y4 + x12 y3 + x14 y2 + x16 y1,
f27 = −x1 y13 − x2 y11 − x3 y9 − x4 y7 − x5 y6 − x6 y5 − x7 y4 − x9 y3 − x11 y2 − x13 y1,
Moreover we put
27∑
i=1
xi fi = 2
27∑
i=1
yi gi
where the quadratic forms gi(x1, . . . , x27)(i = 1, . . . , 27) are given by
g1 = −x13x27 + x16x26 − x18x25 + x20x24 − x22x23,
g2 = −x11x27 + x14x26 − x17x25 + x19x24 − x21x22,
g3 = −x9x27 + x12x26 − x15x25 + x19x23 − x20x21,
g4 = −x7x27 + x10x26 − x15x24 + x17x23 − x18x21,
g5 = −x6x27 − x10x25 + x12x24 − x14x23 + x16x21,
g6 = −x5x27 + x8x26 − x15x22 + x17x20 − x18x19,
g7 = −x4x27 + x8x25 − x12x22 + x14x20 − x16x19,
g8 = x6x26 + x7x25 − x9x24 + x11x23 − x13x21,
g9 = −x3x27 − x8x24 + x10x22 − x14x18 + x16x17,
g10 = x4x26 − x5x25 + x9x22 − x11x20 + x13x19,
g11 = −x2x27 + x8x23 − x10x20 + x12x18 − x15x16,
g12 = x3x26 + x5x24 + x7x20 − x9x18 + x13x15,
g13 = −x1x27 − x8x21 + x10x19 − x12x17 + x14x15,
g14 = x2x26 − x5x23 + x7x20 − x9x18 + x13x15,
g15 = −x3x25 − x4x24 − x6x22 − x11x16 + x13x14,
g16 = x1x26 + x5x21 − x7x19 + x9x17 − x11x15,
g17 = −x2x25 + x4x23 + x6x20 + x9x16 − x12x13,
g18 = −x1x25 − x4x21 − x6x19 − x9x14 + x11x12,
g19 = x2x24 + x3x23 − x6x18 − x7x16 + x10x13,
g20 = x1x24 − x3x21 + x6x17 + x7x14 − x10x11,
g21 = −x2x22 − x3x20 − x4x18 + x5x16 − x8x13,
g22 = −x1x23 − x2x21 − x6x15 − x7x12 + x9x10,
g23 = −x1x22 + x3x19 + x4x17 − x5x14 + x8x11,
g24 = x1x20 + x2x19 − x4x15 + x5x12 − x8x9,
g25 = −x1x18 − x2x17 − x3x15 − x5x10 + x7x8,
24
g26 = x1x16 + x2x14 + x3x12 + x4x10 + x6x8,
g27 = −x1x13 − x2x11 − x3x9 − x4x7 − x5x6
Then we have
27∑
i=1
fi dxi =
27∑
i=1
yi dgi,
27∑
i=1
gi dxi = dg, 3g =
27∑
i=1
xi gi,
where g(x1, . . . , x27) is the cubic form given by
g =− x1 x13 x27 + x1 x16 x26 − x1 x18 x25 + x1 x20 x24 − x1 x22 x23 − x2 x11 x27. + x2 x14 x26
− x2 x17 x25 + x2 x19 x24 − x2 x21 x22 − x3 x9 x27 + x3 x12 x26 − x3 x15 x25 + x3 x19 x23
− x3 x20 x21 − x4 x7 x27 + x4 x10 x26 − x4 x15 x24 + x4 x17 x23 − x4 x18 x21 − x5 x6 x27.
− x5 x10 x25 + x5 x12 x24 − x5 x14 x23 + x5 x16 x21 + x6 x8 x26 − x6 x15 x22
+ x6 x17 x20 − x6 x18 x19 + x7 x8 x25 − x7 x12 x22 + x7 x14 x20 − x7 x16 x19
− x8 x9 x24 + x8 x11 x23 − x8 x13 x21. + x9 x10 x22 − x9 x14 x18 + x9 x16 x17
− x10 x11 x20 + x10 x13 x19 + x11 x12 x18 − x11 x15 x16 − x12 x13 x17 + x13 x14 x15
Thus, by (4.1), we obtain
ω1 = dP2 + ĝ1dX1 − (−a27dX14 + a26dX17 − a25dX19 + a24dX21 − a23dX23
− a22dX24 + a20dX25 − a18dX26 + a16dX27 − a13dX28),
ω2 = dP3 + ĝ2dX1 − (−a27dX12 + a26dX15 − a25dX18 + a24dX20 − a22dX22
− a21dX23 + a19dX25 − a17dX26 + a14dX27 − a11dX28),
ω3 = dP4 + ĝ3dX1 − (−a27dX10 + a26dX13 − a25dX16 + a23dX20 − a21dX21
− a20dX22 + a19dX24 − a15dX26 + a12dX27 − a9dX28),
ω4 = dP5 + ĝ4dX1 − (−a27dX8 + a26dX11 − a24dX16 + a23dX18 − a21dX19
− a18dX22 + a17X24 − a15dX25 + a10dX27 − a7dX28),
ω5 = dP6 + ĝ5dX1 − (−a27dX7 − a25dX11 + a24dX13 − a23dX15 + a21dX17
+ a16dX22 − a14dX24 + a12dX25 − a10dX26 − a6dX28),
ω6 = dP7 + ĝ6dX1 − (−a27dX6 + a26dX9 − a22dX16 + a20dX18 − a19dX19
− a18dX20 + a17dX21 − a15dX23 + a8dX27 − a5dX28),
ω7 = dP8 + ĝ7dX1 − (−a27dX5 + a25dX9 − a22dX13 + a20dX15 − a19dX17
− a16dX20 + a14dX21 − a12dX23 + a8dX26 − a4dX28),
ω8 = dP9 + ĝ8dX1 − (a26dX7 + a25dX8 − a24dX10 + a23dX12 − a21dX14
− a13dX22 + a11dX24 − a9dX25 + a7dX26 + a6dX27),
ω9 = dP10 + ĝ9dX1 − (−a27dX4 − a24dX9 + a22dX11 − a18dX15 + a17dX17
+ a16dX18 − a14dX19 + a10dX23 − a8dX25 − a3dX28),
ω10 = dP11 + ĝ10dX1 − (a26dX5 − a25dX6 + a22dX10 − a20dX12 + a19dX14
+ a13dX20 − a11dX21 + a9dX23 − a5dX26 + a4dX27),
25
ω11 = dP12 + ĝ11dX1 − (−a27dX3 + a23dX9 − a20dX11 + a18dX13 − a16dX16
− a15dX17 + a12dX19 − a10dX21 + a8dX24 − a2dX28),
ω12 = dP13 + ĝ12dX1 − (a26dX4 + a24dX6 − a22dX8 + a18dX12 − a17dX14
− a13dX18 + a11dX19 − a7dX23 + a5dX25 + a3dX27),
ω13 = dP14 + ĝ13dX1 − (−a27dX2 − a21dX9 + a19dX11 − a17dX13 + a15dX15
+ a14dX16 − a12dX18 + a10dX20 − a8dX22 − a1dX28),
ω14 = dP15 + ĝ14dX1 − (a26dX3 − a23dX6 + a20dX8 − a18dX10 + a15dX14
+ a13dX16 − a9dX19 + a7dX21 − a5dX24 + a2dX27),
ω15 = dP16 + ĝ15dX1 − (−a25dX4 − a24dX5 − a22dX7 − a16dX12 + a14dX14
+ a13dX15 − a11dX17 − a6dX23 − a4dX25 − a3dX26),
ω16 = dP17 + ĝ16dX1 − (a26dX2 + a21dX6 − a19dX8 + a17dX10 − a15dX12
− a11dX16 + a9dX18 − a7dX20 + a5dX22 + a1dX27),
ω17 = dP18 + ĝ17dX1 − (−a25dX3 + a23dX5 + a20dX7 + a16dX10 − a13dX13
− a12dX14 + a9dX17 + a6dX21 + a4dX24 − a2dX26),
ω18 = dP19 + ĝ18dX1 − (−a25dX2 − a21dX5 − a19dX7 − a14dX10 + a12dX12
+ a11dX13 − a9dX15 − a6dX20 − a4dX22 − a1dX26),
ω19 = dP20 + ĝ19dX1 − (a24dX3 + a23dX4 − a18dX7 − a16dX8 + a13dX11
+ a10dX14 − a7dX17 − a6dX19 + a3dX24 + a2dX25),
ω20 = dP21 + ĝ20dX1 − (a24dX2 − a21dX4 + a17dX7 + a14dX8 − a11dX11
− a10dX12 + a7dX15 + a6dX18 − a3dX22 + a1dX25),
ω21 = dP22 + ĝ21dX1 − (−a22dX3 − a20dX4 − a18dX5 + a16dX6 − a13dX9
− a8dX14 + a5dX17 − a4dX19 − a3dX21 − a2dX23),
ω22 = dP23 + ĝ22dX1 − (−a23dX2 − a21dX3 − a15dX7 − a12dX8 + a10dX10
+ a9dX11 − a7dX13 − a6dX16 − a2dX22 − a1dX24),
ω23 = dP24 + ĝ23dX1 − (−a22dX2 + a19dX4 + a17dX5 − a14dX6 + a11dX9
+ a8dX12 − a5dX15 + a4dX18 + a3dX20 − a1dX23),
ω24 = dP25 + ĝ24dX1 − (a20dX2 + a19dX3 − a15dX5 + a12dX6 − a9dX9
− a8dX10 + a5dX13 − a4dX16 + a2dX20 + a1dX21),
ω25 = dP26 + ĝ25dX1 − (−a18dX2 − a17dX3 − a15dX4 − a10dX6 + a8dX8
+ a7dX9 − a5dX11 − a3dX16 − a2dX18 − a1dX19),
ω26 = dP27 + ĝ26dX1 − (a16dX2 + a14dX3 + a12dX4 + a10dX5 + a8dX7
+ a6X9 + a4dX11 + a3dX13 + a2dX15 + a1dX17),
ω27 = dP28 + ĝ27dX1 − (−a13dX2 − a11dX3 − a9dX4 − a7dX5 − a6dX6
− a5dX7 − a4dX8 − a3dX10 − a2dX12 − a1dX14),
ϖ2 = a1ω1 + · · ·+ a27ω27 + dP1 − 2ĝdX1 + ĝ1dX2 + · · ·+ ĝ27dX28.
This implies R(X) is given by the following 379 equations;
−P14,28 = P17,27 = −P19,26 = P21,25 = −P23,24 (= a1),
26
−P12,28 = P15,27 = −P18,26 = P20,25 = −P22,23 (= a2),
−P10,28 = P13,27 = −P16,26 = P20,24 = −P21,22 (= a3),
−P8,28 = P11,27 = −P16,25 = P18,24 = −P19,22 (= a4),
−P7,28 = −P11,26 = P13,25 = −P15,24 = P17,22 (= a5),
−P6,28 = P9,27 = −P16,23 = P18,21 = −P19,20 (= a6),
−P5,28 = P9,26 = −P13,23 = P15,21 = −P17,20 (= a7),
P7,27 = P8,26 = −P10,25 = P12,24 = −P14,22 (= a8),
−P4,28 = −P9,25 = P11,23 = −P15,19 = P17,18 (= a9),
P5,27 = −P6,26 = P10,23 = −P12,21 = P14,20 (= a10),
−P3,28 = P9,24 = −P11,21 = P13,19 = −P16,17 (= a11),
P4,27 = P6,25 = −P8,23 = P12,19 = −P14,18 (= a12),
−P2,28 = −P9,22 = P11,20 = −P13,18 = P15,16 (= a13),
P3,27 = −P6,24 = P8,21 = −P10,19 = P14,16 (= a14),
−P4,25 = −P5,25 = −P7,23 = −P12,17 = P14,15 (= a15),
P2,27 = P6,22 = −P8,20 = P10,18 = −P12,16 (= a16),
−P3,26 = P5,24 = P7,21 = P10,17 = −P13,14 (= a17),
−P2,26 = −P5,22 = −P7,20 = −P10,15 = P12,13 (= a18),
P3,25 = P4,24 = −P7,19 = −P8,17 = P11,14 (= a19),
P2,25 = −P4,22 = P7,18 = P8,15 = −P11,12 (= a20),
−P3,23 = −P4,21 = −P5,19 = P6,17 = −P9,14 (= a21),
−P2,24 = −P3,22 = −P7,16 = −P8,13 = P10,11 (= a22),
−P2,23 = P4,20 = P5,18 = −P6,15 = P9,12 (= a23),
P2,21 = P3,20 = −P5,16 = P6,13 = −P9,10 (= a24),
−P2,19 = −P3,18 = −P4,16 = −P6,11 = P8,9 (= a25),
P2,17 = P3,15 = P4,13 = P5,11 = P7,9 (= a26),
−P2,14 = −P3,12 = −P4,10 = −P5,8 = −P6,7 (= a27),
P2,2 = P2,3 = P2,4 = P2,5 = P2,6 = P2,7 = P2,8 = P2,9 = P2,10 = P2,11 = P2,12 = P2,13 = 0,
P2,15 = P2,16 = P2,18 = P2,20 = P2,22 = P3,3 = P3,4 = P3,5 = P3,6 = P3,7 = P3,8 = P3,9 = 0,
P3,10 = P3,11 = P3,13 = P3,14 = P3,16 = P3,17 = P3,19 = P3,21 = P3,24 = P4,4 = P4,5 = P4,6 = 0,
P4,7 = P4,8 = P4,9 = P4,11 = P4,12 = P4,14 = P4,15 = P4,17 = P4,18 = P4,19 = P4,23 = P4,25 = 0,
P5,5 = P5,6 = P5,7 = P5,9 = P5,10 = P5,12 = P5,13 = P5,14 = P5,15 = P5,17 = P5,20 = P5,21 = 0,
P5,23 = P5,26 = P6,6 = P6,8 = P6,9 = P6,10 = P6,12 = P6,14 = P6,16 = P6,18 = P6,19 = P6,20 = 0,
P6,21 = P6,23 = P6,27 = P7,7 = P7,8 = P7,10 = P7,11 = P7,12 = P7,13 = P7,14 = P7,15 = P7,17 = 0,
P7,22 = P7,24 = P7,25 = P7,26 = P8,8 = P8,10 = P8,11 = P8,12 = P8,14 = P8,16 = P8,18 = P8,19 = 0,
P8,22 = P8,24 = P8,25 = P8,27 = P99 = P9,11 = P9,13 = P9,15 = P9,16 = P9,17 = P9,18 = P9,19 = 0,
P9,20 = P9,21 = P9,23 = P9,28 = P10,10 = P10,12 = P10,13 = P10,14 = P10,16 = P10,20 = P10,21 = 0,
P10,22 = P10,24 = P10,26 = P10,27 = P11,11 = P11,13 = P11,15 = P11,16 = P11,17 = P11,18 = 0,
P11,19 = P11,22 = P11,24 = P11,25 = P11,28 = P12,12 = P12,14 = P12,15 = P12,18 = P12,20 = 0,
P12,22 = P12,23 = P12,25 = P12,26 = P12,27 = P13,13 = P13,15 = P13,16 = P13,17 = P13,20 = 0,
P13,21 = P13,22 = P13,24 = P13,26 = P13,28 = P14,14 = P14,17 = P14,19 = P14,21 = P14,23 = 0,
P14,24 = P14,25 = P14,26 = P14,27 = P15,15 = P15,17 = P15,18 = P15,20 = P15,22 = P15,23 = 0,
27
P15,25 = P15,26 = P15,28 = P16,16 = P16,18 = P16,19 = P16,20 = P16,21 = P16,22 = P16,24 = 0,
P16,27 = P16,28 = P17,17 = P17,19 = P17,21 = P17,23 = P17,24 = P17,25 = P17,26 = P17,28 = 0,
P18,18 = P18,19 = P18,20 = P18,22 = P18,23 = P18,25 = P18,27 = P18,28 = P19,19 = 0,
P19,21 = P19,23 = P19,24 = P19,25 = P19,27 = P19,28 = P20,20 = P20,21 = P20,22 = 0,
P20,23 = P20,26 = P20,27 = P20,28 = P21,21 = P21,23 = P21,24 = P21,26 = P21,27 = 0,
P21,28 = P22,22 = P22,24 = P22,25 = P22,26 = P22,27 = P22,28 = P23,23 = P23,25 = 0,
P23,26 = P23,27 = P23,28 = P24,24 = P24,25 = P24,26 = P24,27 = P24,28 = P25,25 = 0,
P25,26 = P25,27 = P25,28 = P26,26 = P26,27 = P26,28 = P27,27 = P27,28 = P28,28 = 0,
P1,28 = P2,28P14,28 + P3,28P12,28 + P4,28P10,28 + P5,28P8,28 + P6,28P7,28,
P1,27 = P2,27P14,28 + P3,27P12,28 + P4,27P10,28 + P5,27P8,28 + P6,28P7,27,
P1,26 = P2,26P14,28 + P3,26P12,28 + P4,25P10,28 − P5,27P7,28 + P5,28P7,27,
P1,25 = P2,25P14,28 + P3,25P12,28 + P4,25P8,28 + P4,27P7,28 − P4,28P7,27,
P1,24 = P2,24P14,28 + P3,25P10,28 − P3,26P8,28 − P3,27P7,28 + P3,28P7,27,
P1,23 = P2,23P14,28 + P3,23P12,28 + P4,25P6,28 − P4,27P5,28 + P4,28P5,27,
P1,22 = P2,24P12,28 − P2,25P10,28 + P2,26P8,28 + P2,27P7,28 − P2,28P7,27,
P1,21 = P2,21P14,28 + P3,23P10,28 − P3,26P6,28 + P3,27P5,28 − P3,28P5,27,
P1,20 = P2,21P12,28 − P2,23P10,28 + P2,26P6,28 − P2,27P5,28 + P2,28P5,27,
P1,19 = P2,19P14,28 + P3,23P8,28 − P3,25P10,28 − P3,27P4,28 + P3,28P4,27,
P1,18 = P2,19P12,28 − P2,23P8,28 + P2,25P6,28 + P2,27P4,28 − P2,28P4,27,
P1,17 = P2,17P14,28 − P3,23P7,28 − P3,25P5,28 − P3,26P4,28 + P3,28P4,25,
P1,16 = P2,19P10,28 − P2,21P8,28 + P2,24P6,28 − P2,27P3,28 + P2,28P3,27,
P1,15 = P2,17P12,28 + P2,23P7,28 + P2,25P5,28 + P2,26P4,28 − P2,28P4,25,
P1,14 = P2,14P14,28 − P3,23P7,27 − P3,25P5,27 − P3,26P4,27 + P3,27P4,25,
P1,13 = P2,17P10,28 + P2,21P7,28 + P2,25P5,28 + P2,26P4,28 − P2,28P4,25,
P1,12 = P2,14P12,28 + P2,23P7,27 + P2,25P5,27 + P2,26P4,27 − P2,27P4,25,
P1,11 = P2,17P8,28 + P2,19P7,28 − P2,24P4,28 − P2,25P3,28 + P3,25P2,28,
P1,10 = P2,14P10,28 + P2,21P7,27 + P2,24P5,27 − P2,26P3,27 + P2,27P3,26,
P1,9 = P2,17P6,28 − P2,19P5,28 − P2,21P4,28 − P2,23P3,28 + P2,28P3,23,
P1,8 = P2,14P8,28 + P2,19P7,27 − P2,14P7,27 − P2,25P3,27 + P2,27P3,25,
P1,7 = P2,14P7,28 − P2,17P7,27 + P2,24P4,25 + P2,25P3,26 − P2,26P3,25,
P1,6 = P2,14P6,28 − P2,19P5,27 − P2,21P4,27 − P2,23P3,27 + P2,27P3,23,
P1,5 = P2,14P5,28 − P2,17P5,27 − P2,21P4,25 − P2,23P3,26 + P2,26P3,23,
P1,4 = P2,14P4,28 − P2,17P4,27 + P2,19P4,25 + P2,23P3,25 − P2,25P3,23,
P1,3 = P2,14P3,28 − P2,17P3,27 − P2,19P3,26 − P2,21P3,25 + P2,24P3,23,
P1,2 = P2,14P2,28 − P2,17P2,27 + P2,19P2,26 − P2,21P2,25 + P2,23P2,24,
28
P1,1 = 2(P2,14P2,28P14,28 − P2,17P2,27P14,28 + P2,19P2,26P14,28 − P2,21P2,25P14,28 + P2,23P2,24P14,28
+P2,14P3,28P12,28 − P2,17P3,27P12,28 + P2,19P3,26P12,28 − P2,21P3,25P12,28 + P2,24P3,23P12,28
+P2,1P4,28P10,28 − P2,17P4,27P10,28 + P2,19P4,25P10,28 + P2,23P3,25P10,28 − P2,25P3,23P10,28
+P2,14P5,28P8,28 − P2,17P5,27P8,28 − P2,21P4,25P8,28 − P2,23P3,26P8,28 + P2,26P3,3,23P8,28
+P2,14P6,28P7,28 − P2,19P5,27P7,28 − P2,21P4,27P7,28 − P2,23P3,27P7,28 + P2,27P3,23P7,28
−P2,17P6,28P7,27 + P2,24P4,25P6,28 + P2,25P3,26P6,28 − P2,26P3,25P6,28 + P2,19P5,28P7,27
−P2,24P4,27P5,28 − P2,25P3,27P5,28 + P2,27P3,25P5,28 + P2,21P4,28P7,27 + P2,23P3,28P7,27
−P2,28P3,23P7,27 + P2,24P4,28P5,27 − P2,26P3,27P4,28 + P2,27P3,26P4,28 + P2,25P3,28P5,27
−P2,28P3,25P5,27 + P2,26P3,28P4,27 − P2,27P3,28P4,25 − P2,28P3,26P4,27 + P2,28P3,27P4,25)
in coordinates (X1, . . . , X28, Z, P1, . . . , P28, P1,1, . . . , P28,28) of L(J).
5. Goursat Equation (Bℓ).
Now, utilizing the Second Reduction Theorem , we will construct the model equation (Bℓ)
from the same standard differential system (X,D) constructed in §3 and §4, which is the local
model corresponding to the Gourst gradation of type Xℓ. First (W ;C
∗, N) is constructed as
follows; W = W (X) is the collection of hyperplanes v in each tangent space Tx(X) at x ∈ X
which contains the fibre ∂D(x) of the derived system ∂D of D, which is same as R(X)
W (X) =
∪
x∈X
Wx ⊂ J(X, 3s+ 1),
Wx = {v ∈ Gr(Tx(X), 3s+ 1) | v ⊃ ∂D(x)} ∼= P1,
Moreover C∗ is the canonical system obtained by the Grassmaniann construction and N is the
lift of ∂D. Precisely, C∗ and N are given by
C∗(v) = ν−1∗ (v) ⊃ N(v) = ν−1∗ (∂D(x)),
for each v ∈ W (X) and x = ν(v), where ν : W (X) → X is the projection.
We introduce a fibre coordinate λ by ϖ = ϖ1 + λϖ2, where
C∗ = {ϖ = 0 } and ∂D = {ϖ1 = ϖ2 = 0}.
Here (w1, w2, z1, . . . , zs, y1, . . . , ys, x1, . . . , xs, λ) constitutes a coordinate system onW (X). Then
we have
dϖ = (dy1 + λdx1) ∧ ω1 + · · ·+ (dys + λdxs) ∧ ωs + dλ ∧ϖ2,
Ch (C∗) = { ϖ = ϖ2 = ω1 = · · · = ωs = dy1 + λdx1 = · · · = dys + λdxs = dλ = 0 },
N = { ϖ = ϖ2 = 0 },
Ch (N) = { ϖ1 = ϖ2 = ω1 = · · · = ωs = dy1 = · · · = dys = dx1 = · · · = dxs = 0 }.
Hence (W (X);C∗, N) is an IG-manifold of corank 1 (see §2.2 [15]). Then we can utilize the
integration of ϖ in §3 and §4. In particular, by the calculation in (3.3) and (4.1), we obtain
(5.1)

ϖ = dZ −
s+1∑
i=1
PidXi,
ϖ2 = dP1 +
s∑
k=1
akdPk+1 + ĝdX1 −
s∑
k=1
ĝkdXk+1.
29
where (X1, . . . , Xs+1, Z, P1, . . . , Ps+1) constitute a coordinate system of J = W/Ch (C
∗) and
(X1, . . . , Xs+1, Z, P1, . . . , Ps+1, a1, . . . , as) constitute a coordinate system of W . Here we note,
in the case of BDℓ type,
f1 =
p+1∑
k=2
xkyk, fk = xky1 + x1yk (k = 2, . . . , p+ 1),
so that
ĝ1 =
1
2
p+1∑
k=2
a2k, ĝk = a1ak (k = 2, . . . , p+ 1), and ĝ =
1
2
p+1∑
k=2
a1a
2
k.
Now our model equation is constructted as follows; Let (R(X);D1X , D
2
X) = (R(W );D
1
W , D
2
W )
be the Lagrange Grassmann bundle over (W ;C∗, N).
R(W ) =
∪
x∈W
Rw, Rw = {v̂ ⊂ N(w) | dϖ |v̂= 0, v̂ is maximal},
where C∗ = {ϖ = 0}. Moreover D1W and D2W are defined by
D1W (v̂) = τ
−1
∗ (C
∗(w)) ⊃ D2W (v̂) = τ−1∗ (v̂),
for v̂ ∈ R(W ), w = q(v̂) and τ : R(W ) → W is the projection. Namely we collect maximal
isotropic subspaces of (N(w), dϖ). Infact v = q∗(v̂) is a legendrian subspace of (J,C) such that
v ⊂ ι(w) = q∗(N(w)), where q : W → J = W/Ch (C∗) is the projection and ι : W → I1(J) is
the canonical immersion (see Theorem 2.1 [15]). Thus we define a map ζ : R(W ) → L(J) by
ζ(v̂) = q∗(v̂). Then we have
ζ(Rw) = {v ∈ L(J) | v ⊂ w̄ ⊂ C(u)} ∼= L(w̄/w̄⊥) ∼= U(s)/O(s),
where u = q(w), w̄ = ι(w) and L(w̄/w̄⊥) denotes the Lagrange Grassmann manifold of the
symplectic vector space w̄/w̄⊥of dimension 2s. Hence our equation ζ(R(W )) is the collection of
legendrian subspaces v = q∗(v̂) such that v ⊂ w̄ = q∗(N(w)), i.e. v̂ ⊂ N(w), for w ∈ W . Since
(X, ∂D) is a regular differential system of type c1(s, 2) and of Cartan rank s, we can check that
(R(X);D1X , D
2
X) is a PD manifold of second order on an open dense subset R̂ of R(X) = R(W )
(see Proposition 7.3 [15] for detail) and ζ : R(W ) → L(J) is an immersion on R̂.
Now, substituting dPi −
∑s+1
j=1 PijdXj (i = 1, . . . , s+1) into (5.1) , we obtain the parametric
description of Goursat equation (Bℓ)
(5.2) P11 +
s∑
k=1
akPk+1,1 + ĝ = 0, P1,j+1 +
s∑
k=1
akPk+1,j+1 − ĝj = 0 (j = 1, . . . , s).
In fact we can describe the immersion ζ : R(W ) → L(J) in coordinates as follows (see §4.1
[15] for detail); Let (X1, Xi+1, Z, P1, Pi+1, A
i+1, Bi+1, S)(1 ≦ i ≦ s) be the coordinate system of
I1(J) induced from a canonical coordinate (X1, . . . , Xs+1, Z, P1, . . . , Ps+1) of J (see §2.1 [15]).
Then, by (5.1), ι(W ) is given by
(5.3) Ai+1 = −ai, Bi+1 = ĝi, S = −ĝ +
s∑
i=1
Ai+1Bi+1 = −4ĝ.
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Moreover, following the argument in §4.1 [15], we can choose the cooodinate system (X1, Xi+1, Z,
P1, Pi+1, ai, P
∗
i+1,j+1) (1 ≦ i, j ≦ s) of R(W ) so that ζ : R(W ) → L(J) is given by
ζ∗Pi+1,j+1 = P
∗
i+1,j+1, ζ
∗P1,i+1 = ι
∗Bi+1 +
s∑
j=1
P ∗i+1,j+1ι
∗Aj+1,
ζ∗P11 = ι
∗S +
s∑
i=1
s∑
j=1
P ∗i+1,j+1ι
∗Ai+1ι∗Aj+1,
where (X1, Xi+1, Z, P1, Pi+1, P11, P1,j+1, Pi+1,j+1)(1 ≦ i, j ≦ s) is the coordinate system of L(J)
induced from a canonical coordinate system (X1, Xi+1, Z, P1, Pi+1)(1 ≦ i ≦ s) of J .
Then, substituting (5.3) into the above, we obtain
ζ∗P1,i+1 = ĝi −
s∑
j=1
ζ∗Pi+1,j+1aj,
ζ∗P11 = −4ĝ +
s∑
i=1
s∑
j=1
ζ∗Pi+1,j+1aiaj = −4ĝ −
s∑
i=1
(ζ∗P1,i+1 − ĝi)ai = −
s∑
i=1
ζ∗P1,i+1ai − ĝ.
This gives us (5.2).
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K.Yamaguchi, Department of Mathematics,, Faculty of Science,, Hokkaido University,, Sap-
poro 060-0810,, Japan, E-mail yamaguch@math.sci.hokudai.ac.jp
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