Homogeneous Real (2,3,5) Distributions with Isotropy

We classify multiply transitive homogeneous real (2,3,5) distributions up to local diffeomorphism equivalence.


Introduction
The study of (2,3,5) distributions, that is, tangent 2-plane distributions (M, D) on 5-manifolds satisfying the genericity condition [D, [D, D]] = T M , dates to Cartan's so-called "Five Variables Paper" [8]. That article (1) resolved the equivalence problem for these geometries and in doing so revealed a surprising connection between these geometries and the exceptional complex Lie algebra of type G 2 , and (2) (nearly) locally classified complex (2,3,5) distributions D whose infinitesimal symmetry algebra aut(D) has dimension at least 6. Besides its historical significance and the connection with G 2 , which mediates its relationship with other geometries [17], [22, § 5], this geometry is significant in part because of its appearance in simple, nonholonomic kinematic systems [4,5], and it has enjoyed heightened attention in the last decade or so, owing in part to its realization in the class of parabolic geometries [6, § 4.3.2], a wide class of Cartan geometries for which many powerful results are available.
In the current article we classify (again, locally) all homogeneous real (2, 3, 5) distributions with multiply transitive symmetry algebra, so again those for which dim aut(D) ≥ 6. Our motivation for carrying out this classification is twofold: First, it gives a canonical list of examples that additionally have favorable symmetry properties for work in progress about real (2,3,5) distributions. But it is also independently interesting, in part because of the appearance of several distinguished rolling distributions (see § 7).
Our method is straightforward and is analogous to that, e.g., of [13]: Any multiply transitive homogenous (2, 3, 5)-distribution (M, D) can be encoded in an algebraic model (h, k; d) in the sense that the original distribution can (up to local diffeomorphism equivalence) be recovered and hence specified by the model data. Here, h := aut(D) is the (infinitesimal) symmetry algebra of D, k is the isotropy subalgebra of a point u ∈ M (again, by hypothesis dim k ≥ 1), and d ⊂ h is the subspace corresponding to D u . Given any real algebraic model, its complexification (h ⊗ C, k ⊗ C; d ⊗ C) is a complex algebraic model, and conversely the given real algebraic model can be recovered from an appropriate antiïnvolution φ : h ⊗ C → h ⊗ C admissible in the sense that it preserves the filtration k ⊗ C ⊂ d ⊗ C ⊂ h ⊗ C.
We thus briefly recall in § 4 (and summarize in Table 1 in Appendix A) the classification of complex (2,3,5) distributions. For each distribution in the classification we give an explicit algebraic model in terms of abstract Lie algebra data. Most of these distributions were identified by Cartan [8], but Doubrov and Govorov found (much) later that Cartan's list omitted the model we here call N.6 [10].
Then, in § 5, for each of the complex algebraic models (h, k; d) we produced in § 4 with dim h = 6 we classify the admissible antiïnvolutions of h up to a notion of equivalence that corresponds to diffeomorphism equivalence of the homogeneous distributions. Together with the real algebraic model O R with maximal symmetry (unique up to equivalence) and the preexisting classification of the models with dim h = 7 [14, Theorem 2] (which are here denoted N.7 R r,s ), this yields the main result of this article: Theorem A. Any multiply transitive homogeneous real (2,3,5) distribution is locally equivalent to exactly one distribution in Table 2, modulo the given equivalences of parameters.
In § 6 we give algorithms for identifying, in both the complex and real cases, a multiply transitive homgeneous (2,3,5) distribution given in terms of an abstract algebraic model among the distributions in the classification; this amounts to constructing sufficiently many invariants to distinguish all of the models. Finally, in § 7 we identify many of the distributions in the real classificiation as rolling distributions, that is, 2-plane distributions on 5-manifolds defined on the configuration space of two surfaces rolling along one another by the kinematic no-slip and no-twist conditions, some of which are already well-known.
It is a pleasure to thank Boris Doubrov, both for several helpful conversations and for access to the unpublished notes [11]. It is likewise a pleasure to thank Dennis The for an invaluable exchange about the classification of real forms of a given complex geometric structure. The author gratefully acknowledges support from the Austrian Science Fund (FWF), via project P27072-N25, the Simons Foundation, via grant 346300 for IMPAN, and the Polish Government, via the matching 2015-2019 Polish MNiSW fund.
2. The geometry of (2, 3, 5) distributions  [D, D]] is the condition that [D, D] has constant rank; for a (2,3,5) distribution, rank[D, D] = 3. We will work in both the smooth and complex categories. In both cases we will always assume that M is connected.
2.1. Monge (quasi-)normal form. One can construct 2-plane distributions via ordinary differential equations of the form (1) z ′ (x) = F (x, y, y ′ , y ′′ , z) : We can prolong any solution (x, y(x), z(x)) to a curve (x, y(x), y ′ (x), y ′′ (x), z ′ (x)) in the (partial) jet space J 2,0 (F, F 2 ) ∼ = F 5 (F = R or F = C), and by construction any such curve is tangent to the 2-plane distribution D F ⊂ T F 5 defined in the respective jet coordinates (x, y, p, q, z) as the common kernel of the canonical jet 1-forms dy − p dx and dp − q dx and the 1-form dz − F (x, y, p, q, z) dx defined by (1). Conversely, the projection of any integral curve of this distribution to which the pullback of dx vanishes nowhere defines a solution of this o.d.e. The distribution D F is spanned by ∂ q and ∂ x + p∂ y + q∂ p + F (x, y, p, q, z)∂ z -the latter is the total derivative-and computing directly shows that D F is a (2, 3, 5) distribution iff ∂ 2 q F vanishes nowhere. Goursat showed that, in fact, every (2, 3, 5) distribution arises locally this way and hence can be specified by some function F of five variables. Such an o.d.e. (or, by slight abuse of terminology, the function F itself) is called a Monge normal form of the distribution.

Homogeneous distributions
3.1. Infinitesimal symmetries. An infinitesimal symmetry of a (2, 3, 5) distribution (M, D) is a vector field ξ ∈ Γ(D) whose (local) flow preserves D, or equivalently for which L ξ η ∈ Γ(D) for all η ∈ Γ(D). We denote the set of infinitesimal symmetries, called the (infinitesimal) symmetry algebra by aut(D), and we say that (M, D) is infinitesimally homogeneous if aut(D) is transitive, that is, if {ξ u : ξ ∈ aut(D)} = T u M for all u ∈ M . In this article we are concerned with multiply transitive homogeneous distributions, that is, those for which the isotropy subalgebra k u < aut(D) of infinitesimal symmetries vanishing at any u ∈ M is nontrivial, or equivalently, for which dim aut(D) ≥ 6.

3.2.
Algebraic models for homogeneous distributions. Fix a homogeneous (2, 3, 5) (real or complex) distribution (M, D) with transitive symmetry algebra h := aut(D), fix a point u ∈ M , and denote by k < h the subalgebra of vector fields in h vanishing at u and by d ⊂ h the subspace d := {ξ ∈ h : ξ u ∈ D u }. Given an algebraic model, we can reconstruct D up to local equivalence: For any groups H, K respectively realizing h, k and with H > K, invoke the canonical identification by k-invariance this definition is independent of the coset representative h, and by genericity D is an H-invariant (2, 3, 5) distribution. Via the above identification, 1 We declare two algebraic models (h, k; d), (h ′ , k ′ ; d ′ ) to be equivalent iff there is a Lie algebra automorphism α : h → h ′ satisfying α(k) = k ′ and α(d) = d ′ . Unwinding definitions shows that equivalent algebraic models determine locally equivalent distributions.

Classification of multiply transitive homogeneous complex distributions
In his tour de force "Five Variables" paper [8], Cartan showed that for all (2,3,5) distributions D, dim aut(D) ≤ 14, and that equality holds iff it is locally equivalent to the so-called flat distribution ∆; his argument applies to both the real and complex settings. We call the corresponding (complex) algebraic model O (see § 4.1). In this case, aut(D) is isomorphic to the simple complex Lie algebra of type G 2 -we denote it by g 2 (C)-and we say that D is (locally) flat. Cartan furthermore claimed to classify up to local equivalence and (implicitly in the complex setting) all distributions D with dim aut(D) ≥ 6. 2,3 Doubrov and Govorov found (much) later, however, that Cartan's classification misses a single distribution up to local equivalence [10]; this is model N.6 (see § 4.3). See [28] for a short, expository account.
In this section, we summarize the classification of multiply transitive homogeneous complex (2, 3, 5) distributions D up to local equivalence. For each, we give (1) an explicit algebraic model (h, k; d) in terms of abstract Lie algebra data, (2) a complement of d in d + [d, d], (3) a Monge normal form function F realizing the distribution, and (4) an explicit isomorphism h ∼ = aut(D F ).
We use this list of complex algebraic models to classify the real algebraic distributions in § 5. We use the convention that the undecorated Fraktur names g 2 , gl m , sl m , so m , n m refer to real Lie algebras, and we denote their complexifications by gl m (C) and analogously. By mild abuse of notation, for any element v of a real Lie algebra g we also denote by v the element v ⊗ 1 ∈ g ⊗ C.
Remark 3. Our convention for labeling the nonflat distributions D in the classification refers both to the dimension of aut(D) and to a particular discrete invariant. The fundamental curvature quantity of a (2, 3, 5) distribution (M, D) is a section A ∈ Γ(S 4 D * ) [8, ¶ VI § 33], and its nonvanishing is a complete local obstruction to local equivalence to the model O ( § 4.1) [8, ¶ VIII], hence the epithet flat. At each point u ∈ M the Petrov type (or root type) of A u is the multiplicities of the roots of A u ; if D is real, we instead use the multiplicities of the roots of A u ⊗ C. If D is infinitesimally homogeneous, the Petrov type is the same at all points, and among multiply transitive homogeneous distributions only Petrov types D (two double roots), N (a quadruple root), and O (A = 0) occur. 4 4.1. The flat model O. Denote by g 2 the split real form of g 2 (C), take q < g 2 to be the (paraboic) subalgebra of elements fixing an isotropic line in the standard representation of g 2 (cf. [4, § 4]), and denote by q + < q the orthogonal of q with respect to the Killing form on g 2 . Then, define the subspace 1 Conversely, a triple (h, k; d), where h is a Lie algebra, k < h is a Lie subalgebra, and d ⊂ h is a subspace for which h ⊃ k and dim(d/k) = 2, together satisfying the k-invariance and genericity conditions, determines up to local equivalence a homogeneous distribution via this construction. The symmetry algebra of this distribution may be strictly larger that h, however; for example, in (6) dim h = 6, but for the excluded value λ = 9, the resulting distribution ∆ is flat, so dim aut(∆) = dim g 2 = 14. 2 Cartan's classification is restricted to distributions for which the Petrov type of the distribution is the same at all points; this condition holds automatically for locally homogeneous distributions. See Remark 3. 3 Cartan's classification contains a family of distributions with symmetry algebra of dimension 6 whose symmetry algebra is not transitive, namely those in [8, ¶ IX] with I nonconstant.
4 Among (not necessarily multiply) transitive distributions, types I (four single roots) and II (a double root and two single roots) also occur, but type III (a triple root and a single root) does not. [11] g −1 2 := {ξ ∈ g 2 : [q + , ξ] ⊆ q}. The Killing form bracket identity and the Jacobi identity together give [q(C), g 2 (C) −1 ] ⊆ g 2 (C) −1 (in fact, equality holds), and inspecting the root diagram of g 2 -or just using the explicit realization g 2 < gl 7 in Appendix B-gives that dim(g −1 is an algebraic model, O, of the complex flat distribution. Then, Importantly, the flat distribution can also be realized by the Monge normal form function F (x, y, p, q, z) = q 2 , so, by the so-called Hilbert-Cartan equation z ′ (x) = y ′′ (x) 2 [9,19]. In Appendix B we give an explicit isomorphism g 2 ∼ = aut(D F ); it can just as well be regarded as an identification of complex data.
The quantity I 2 ∈ C is a complete invariant. It is convenient for our purposes to use a generalization of this form studied by Doubrov and Zelenko in the context of control theory: Define [14] (3) F r,s (x, y, p, q, z) = q 2 + rp 2 + sy 2 , r, s ∈ C, and denote the distribution it determines by D r,s . Then, D r,s is locally equivalent to the flat model iff the roots of the polynomial t 4 − rt 2 + s form an arithmetic sequence, that is, if 9r 2 = 100s; otherwise it is submaximal. In § 6.1.1 we recover the facts that (1) the distributions D r,s and D r ′ ,s ′ are locally equivalent iff there is a constant c ∈ C−{0} such that r ′ = cr, s ′ = c 2 s, suggesting the invariant J = 4s/r 2 and (2) D r,s has Cartan invariant I 2 = 9r 2 /(100s − 9r 2 ) = 9/(25J − 9). We realize the distributions D r,s as abstract models as follows. Let n 5 = n −2 5 ⊕ n −1 5 be the 5dimensional real Heisenberg algebra endowed with its standard contact grading, and fix a standard basis (u, s 1 , s 2 , t 1 , t 2 ), so that [s 1 , t 1 ] = [s 2 , t 2 ] = u (and all brackets of basis elements not determined by these identities are zero). Let E ∈ Der(n 5 ) denote the grading derivation, so that [E, s a ] = −s a and [E, t a ] = −t a for a = 1, 2 and [E, u] = −2u. For each parameter value (r, s) we choose a derivation F ∈ Der(n 5 (C)) commuting with E and extend n 5 (C) by these derivations to produce the Lie algebras h C r,s := n 5 (C) ⋌ E, F = n 5 (C) ⋌ C 2 occurring in the respective abstract models. In each case, we give a Lie algebra isomorphism h C r,s ∼ = aut(D r,s ) identifying E ↔ y∂ y + p∂ p + q∂ q + 2z∂ z and F ↔ ∂ x and identifying u with a constant multiple of ∂ z .
Following [12, § 3], choose a, b so that r = a 2 + b 2 , s = a 2 b 2 . Then, for µ = 0, (a, b) and (µa, µb) determine equivalent distributions, so we may specify a distribution by [a : b] ∈ P 1 . (The condition that D r,s is not flat is equivalent to the requirement that a = ±3b and b = ±3a.) It is convenient to split cases according to whether s = 0 (equivalently, whether ab = 0) and whether the discriminant r 2 − 4s of the auxiliary polynomial t 2 − rt + s is zero (equivalently, whether b = ±a).
Generic case (s = 0, r 2 − 4s = 0). Define F and then the algebraic model N.7 r,s by 8 This is a special case of a much more general construction [6, § 3] related to the realization of (2, 3, 5) distributions as so-called parabolic geometries [6, § 4.3.2], [23]. 6 As in Footnote 2, the bound in [8] is established for distributions with constant Petrov type; this assumption was removed in [21]. 7 Here the factor 10 3 corrects a numerical error of Cartan, first identified, to the author's knowledge, in [26]. 8 Alternatively one can choose k C r,s and d C r,s here in a way more symmetric in the coefficients of the basis elements s i and t i at the cost of introducing radicals in a and b: Replace the first element in the definition of k C r,s to be √ bs 1 − √ as 2 + √ bt 1 + √ bt 2 and the first element in that of d C r,s by Case s = 0. By relabeling we may assume b = 0 and thus r = a 2 . Define the algebraic model N.7 r,0 by Case r 2 − 4s = 0. Equivalently, s = r 2 /4, or b = ±a (without loss of generality, we take b = a). Define the algebraic model N.7 r,r 2 /4 by Remark 4. One can also realize the submaximal distributions via Monge normal forms F (x, y, p, q, z) = q m , m = 0, 1, (corresponding to r = (2m−1) 2 +1, s = (2m−1) 2 ) and F (x, y, p, q, z) = log q (r = 2, s = 1).

4.3.
The Doubrov-Govorov model N. 6. In [10] Doubrov & Govorov reported finding a homogeneous distribution D with dim aut(D) = 6 missing from Cartan's ostensible classification and gave it in terms of a Monge normal form, F (x, y, p, q, z) := q 1/3 + y. 9 We indicate briefly, using this distribution as an example, how to produce an algebraic model from a locally homogeneous distribution given in local coordinates (for example, in Monge normal form). From [10] the symmetry algebra h = aut(D F ) has basis but we can also compute h with the Maple package DifferentialGeometry using the following routine. The subspace ξ 1 , ξ 2 , ξ 3 is a subalgebra isomorphic to sl 2 (C), and we may identify (ξ 1 , ξ 2 , ξ 3 ) with the complexification of a standard basis (x, y, h) of sl 2 , namely one satisfying [x, y] = h, [h, x] = 2x, [h, y] = −2y. The radical r of h is isomorphic to the complexification of the 3-dimensional real Heisenberg algebra n 3 , and we may identify the basis (f 4 , f 5 , f 6 ) of r with the complexification of a standard basis (u, s, t) 9 Other Monge normal forms for this distribution include F (x, y, p, q, z) = q 2/3 + p 2 [11] and F (x, y, p, q, z) = e y [1 + e −2y/3 (q − 1 2 p 2 ) 1/3 ]; the latter corrects an error in [ At the basepoint u := (0, 0, 0, 1, 0) ∈ C 5 , the isotropy subalgebra is k = ξ 1 , and ∂ q = 1 3 ξ 3 and
One Monge normal form for this distribution is F (x, y, p, q, z) = p −2 q 2 , 12 and we may identify 4.6. The model D.6 * with complex Euclidean symmetry. Realize so 1,2 < gl 3 ∼ = End(R 3 ) concretely as the Lie algebra of endomorphisms preserving the nondegenerate, symmetric bilinear form  The restriction of the standard action of gl 3 on R 3 defines a semidirect product 2 ⋌ R , . If we take (w 1 , w 2 , w 3 ) to be the standard basis of R 3 , the Lie bracket on sl 2 ⋌ R 3 is characterized by and complexifying realizes so One Monge normal form for this distribution is F (x, y, p, q, z) = q log q + p 2 , 13 and we may identify

Classification of mutliply transitive homogeneous real distributions
We now ply the list in § 4 to classify multiply transitive homogeneous real (2, 3, 5) distributions.

Real forms of complex algebraic models. Given a real algebraic model
is a complex algebraic model; we call the latter the complexification of the former. Conversely, suppose that we have a complex algebraic model (h, k; d). Recall that a real form of h is the fixed point algebra h φ of an antiïnvolution φ : h → h, that is, a complex-antilinear map satisfying We say that two admissible antiïnvolutions φ, ψ are equivalent if ψ = α • φ • α −1 for some admissible automorphism α of h, that is, one preserving k and d. Two admissible antiïnvolutions are equivalent iff they determine equivalent real algebraic models (and hence locally equivalent real homogeneous distributions), so to classify the latter one can classify the former. Not all h admit admissible antiïnvolutions, and hence not all complex algebraic models admit real forms. 12 Other Monge normal forms for this distribution include F (x, y, p, q, z) = p −3 q 2 , F (x, y, p, q, z) = q/y (both given in [11]), F (x, y, p, q, z) = [x 2 (qx 2 + px − y)] −1 (this is the special case ǫ = 1 of [26, Example 6.5.3]), and F (x, y, p, q, z) = 1 2 (q − 1 2 p 2 ) 1/2 (the special case m = 0, n = 1 2 of [26, Example 6.7.4]). 13 Another Monge normal form for this distribution is F (x, y, p, q, z) = log q + y [11].
We thus classify the multiply transitive homogeneous real distributions as follows: Up to local equivalence there is only a single real flat distribution and hence only a single real form of O. For the submaximal models we appeal to the existing classification [ so that with respect to (e a ) any admissible antiïnvolution φ of h is commensurately block lower-triangular; in particular, any admissible antiïnvolution φ satisfies φ(e 6 ) = ζe 6 with |ζ| = 1. Any such φ also preserves any other subspaces of h constructed invariantly from the data (h, k; d), and in each case we are able to choose a basis well-adapted to some of these, which restricts further in a convenient way the form of φ with respect to the basis. Next, for any automorphism α : h → h, by definition the constants σ ij := [α(e i ), α(e j )] − α([e i , e j ]) all vanish, and we impose those vanishing conditions to determine the admissible antiïnvolutions φ of the complex algebraic model. After classifying them up to equivalence, we record a representative antiinvolution φ, the corresponding real model (h φ , k φ ; d φ ), and a complement of 5.1.1. Local coordinate realizations. For some purposes it is convenient to have local coordinate expressions of distributions. We give such forms for many of the real models in the classification, in some cases via Monge normal forms, and indicate procedures for producing them in others.
• For each complex algebraic model for which the data specifying the model can be interpreted as real, the first real form identified is the one given by interpreting thus, the Monge normal form function F for the complex distribution can be regarded as one for that real model, and the given explicit isomorphism h ↔ aut(D F ) can be regarded as an identification of the real data. This applies to all models without parameters (all models other than N.7 r,s and D.6 λ ) as well as to N.7 r,s , r, s ≥ 0, r 2 ≥ 4s, and D.6 λ , λ > 0. • The submaximal real models, namely, the real forms N.7 R r,s of the models N.7 r,s were classified in [12], and Monge normal forms were recorded there. See § 5.3.
• Example 5 later in this section outlines, using the real form D.6 4 * of model D.6 * as an example, how to construct local coordinate realizations of a homogeneous distribution. • Example 6 in § 6 applies the algorithm in that section to show that a particular function F defines a Monge normal form for the real form N.6 − of model N.6. • Section 7 realizes several of the real models in the classification as rolling distributions, from which one can readily construct coordinate realizations; this is carried out explicitly for the real forms D.6 6 λ of D.6 λ , λ > 0.
5.2. The flat model. As in the complex case, up to local equivalence there is a unique flat distribution. Thus all admissible antiïnvolutions of g 2 (C) are equivalent; taking complex conjugation gives the model The real flat model can be described as the rolling distribution (see § 7 below) determined by a pair of spheres, one whose radius is thrice that of the other [4], 14 and also as a canonical distribution determined by the algebra O of split octonions on the null quadric in the projectivization P(Im O) of the space of purely imaginary split octonions, [4, § 6], [23]; see also [3].  r,s (x, y, p, q, z) = q 2 + rp 2 + sy 2 for some r, s ∈ R, and as in the complex case, the distribution D r,s so determined is locally equivalent to the flat model iff 9r 2 = 100s, which again we henceforth exclude. Otherwise, D r,s and D r ′ ,s ′ are locally equivalent iff there is a constant c ∈ R + such that r ′ = cr, s ′ = c 2 s.
If (h, k; d) is a submaximal real algebraic model, then it follows from the algorithm in § 6.1.1 that the Cartan invariant I 2 of its complexification in real, so reality of that invariant is a necessary condition for existence of a real form of a complex algebraic model. The form of the equivalence relation implies that the triple (I 2 , sign(r), sign(s)) is a complete invariant of the model.

5.5.
The models with complexified symmetry algebra sl 2 (C) ⊕ sl 2 (C). This case is the most involved. We will see that (1) in some cases the qualitative features of the real forms (including the isomorphism types of the real forms h φ ) depend on the sign of λ, and (2) the case λ = −1 is distinguished. In § 6.2.1 we show that λ is an invariant of the distribution up to inversion, and it is manifestly real for distributions that are complexifications of real distributions. Thus, only the models D.6 λ with λ real can admit real forms, and we therefore restrict to such λ. (Since the expressions defining the models with real λ use only real coefficients, all such models do admit real forms.) Before proceeding, we recall the fact, used below, that the real forms of sl 2 (C) ⊕ sl 2 (C) ∼ = so 4 (C) are characterized up to isomorphism by the signatures of their Killing forms: 15 sl 2 (C) R ∼ = so 1,3 (4, 2) sl 2 ⊕ sl 2 ∼ = so 2,2 .
We will see that all four forms occur in the classification.
Forming a suitable linear combination of the e 3 and e 6 components of σ 15 gives that (λ + 1)(γ − ζ) = 0, so γ = ζ or λ = −1. For each subcase, the conditions σ 16 = σ 26 = 0 imply the vanishing either of both δ 5 and ǫ 4 or of both δ 4 and ǫ 5 , then σ 12 = 0 implies that one of the two remaining quantities can be written in terms of the other, after which all of the conditions σ ij = 0 are satisfied. Case γ = ζ. We split cases according to the sign of γ = ζ = ±1.
Subsubsubcase λ > 0. The Killing form of h φ is definite, so We can realize the above identification via We call this model D.6 6 λ . Exchanging the direct summands so 3 is an isomorphism between D.6 6 λ and D.6 6 1/λ , and this exhausts the isomorphisms among these models. Subsubsubcase λ < 0. The Killing form of h φ has signature (2, 4), so and we can realize this isomorphism by identifying a ⊕ 0, b ⊕ 0, c ⊕ 0 respectively with the elements identified with a, b, c in the λ > 0 case, and identifying Then, The isomorphism sl 2 ⊕ so 3 → so 3 ⊕ sl 2 given by reversing the order of the factors identifies the model with parameter value λ with the model D.6 4 1/λ , so this branch contributes no new models.
Subcase δ = 1. The real form h φ is spanned by {e 1 + e 2 , i(e 1 − e 2 ), ie 3 , e 4 + e 5 , i(e 4 − e 5 ), ie 6 }. We may identify Here we realize so 2 as the Lie algebra preserving the standard inner product on R 2 , and we take z ′ to be its standard generator, so that its action is given by Then, We denote the model by D.6 2 ∞ . Subcase δ = −1. The real form h φ is spanned by {e 1 − e 2 , i(e 1 + e 2 ), ie 3 , e 4 − e 5 , i(e 4 + e 5 ), ie 6 }. We may identify . Then, We denote the model by D.6 4 ∞ .
Subcase k even. In this case, h φ is spanned by {ie 1 , e 2 , ie 3 , e 4 , ie 5 , ie 6 }. We may identify (where the semidirect product is given by the same action as in the case ζ = +1) via x ↔ i(e 1 + 1 2 e 5 + 1 2 e 6 ), in which case We denote the model by D.6 1+ * . Subcase k odd. Here, h φ is the real span of {e 1 , ie 2 , ie 3 , ie 4 , e 5 , ie 6 }. We may identify the real form with the real Euclidean algebra, Take the standard basis (w a , w b , w c ) of R 3 , so that the action is characterized by [a, w b ] = w c , its cyclic permutations in a, b, c, and [a, w a ] = [b, w b ] = [c, w c ] = 0. Then, we may identify in which case We denote the model by D.6 3 * . Example 5. We indicate, using D.6 3 * as an example, how to realize a multiply transitive homogeneous distribution in local coordinates starting from an algebraic model.
In these coordinates the fibers of H → H/K are the integral curves of ∂ ν , so we may use (λ, µ, r, s, t) as coordinates on the quotient space. Pulling back the 1-forms defining d (viewed under this identification as a local distribution on H) by a suitable local section and computing the annihilator gives a coordinate expression for the distribution D: ∂ λ − cos 2 µ ∂ r − sin λ sin µ cos µ ∂ s + cos λ sin µ cos µ ∂ t , ∂ µ − cos λ∂ s − sin λ∂ t .
Following the procedure in [26, § 2] allows us to put this distribution in preferred forms convenient for other purposes. In coordinates (x i ), x 1 = λ, x 2 = s cos λ + t sin λ + µ, x 3 = −s sin λ + t cos λ, x 4 = r, x 5 = tan µ, D is the common kernel of the 1-forms where f := x 5 arctan x 5 − x 2 x 5 + 1; distributions in this form for some function f (x 1 , . . . , x 5 ) are said to be in Goursat normal form. For any distribution in that form, changing to coordinates (x, y, p, q, z), where realizes the distribution in Monge normal form with the function F given by writing x 5 ∂ x 5 f − f in the coordinates (x, y, p, q, z). In our case, ∂ x 5 f = x 5 /[(x 5 ) 2 + 1] + arctan x 5 − x 2 , and so one Monge normal form for D is given by F (x, y, p, q, z) = − cos 2 g −1 (y + q); here g is the map u → u + sin u cos u, which, up to composition with appropriate affine transformations, appears in Kepler's equation and in the standard parameterization of the cycloid.

Identification algorithms
We now present an algorithm for identifying the isomorphism type of a given abstract model (h, k; d) with an explicit model in the classification; this amounts to generating sufficiently many invariants to distinguish different distributions. We split cases according to dim h; up to isomorphism there is only one model with dim h = 14 in both the complex and real settings.
6.1. Models with dim h = 7. In this case, in both the complex and real settings, the models are determined up to equivalence by the underlying Lie algebra h, so it is enough to distinguish those algebras.
6.1.1. Complex models. (Motivated by the discussion in [12, § 2],) consider the maps ad v, v ∈ h. Not all maps ad v are tracefree, so t := {v ∈ h : tr ad v = 0} has dimension 6, but the nilradical n < t is isomorphic to n 5 (C) and so has dimension 5. Thus, the spectrum of ad v • depends only on the projection of v • to the line t/n, and so the conformal spectrum of ad v • is independent of the choice of v • ∈ t − n.
In particular, where σ k denotes the kth symmetric polynomial in the eigenvalues of ad v • , is an invariant of the conformal spectrum of ad v • and hence an invariant of h; here J is normalized to coincide with the invariant of the same name appearing in [20]. For the Lie algebras h C r,s = n 5 (C) ⋌ E, F in the abstract models, the nilradical is n 5 (C) and t = n 5 (C) ⊕ F , so we need only compute the spectrum of ad F. Consulting the formulae in § 4.2 gives that the spectrum is (+a, −a, +b, −b, 0, 0, 0), so σ 2 = −(a 2 + b 2 ) = −r and σ 4 = a 2 b 2 = s, and thus J = 4a 2 b 2 /(a 2 + b 2 ) 2 = 4s/r 2 . Specializing to Cartan's Monge normal form (r = 10 3 I, s = I 2 + 1) gives J = 9(I 2 + 1)/(25I 2 ); thus, since I 2 is a complete invariant, so is J.
6.1.2. Real models. From § 5.3 every submaximal complex model that admits a real form admits precisely two. So, to identify a submaximal real abstract model (h, k; d), we determine the complexification (h ⊗ C, k ⊗ C; d ⊗ C) and then determine which of the two real forms one started with.
If we define t, n as in the complex case, then replacing a choice v • ∈ t − n with µv • + n, µ ∈ R * , n ∈ n, respectively replaces σ k by µ k σ k , so the signs of σ 2 , σ 4 are independent of the choice v • . For real pairs (r, s), the formulae derived in the complex case give for h r,s and the choice v • = F that sign(σ 2 ) = − sign(r) and sign(σ 4 ) = sign(s), so for all (r, s) the pair (sign(σ 2 ), sign(σ 4 )) distinguishes between the real forms.
6.2.1. Complex models. The isomorphism type of h can be determined by examining the radical, r: Consulting Table 1, we see that if r = {0}, then dim r = 3 and the derived algebra [r, r] has dimension 0, 1, or 2, respectively, if h is isomorphic to so 3 ⋌ C 3 , so 3 ⋌ n 3 , or so 3 ⊕ (so 2 ⋌ C 2 ), in which case the model is isomorphic, respectively, to D.6 * , N.6, or D.6 ∞ .
If instead r = {0}, then h ∼ = sl 2 (C) ⊕ sl 2 (C), and the model is isomorphic to D.6 λ for some parameter λ (which, recall, is only defined up to inversion, λ → 1/λ). Let π 1 , π 2 : h → sl 2 (C) denote the projections onto the two summands, and define e := d ∩ k ⊥ . Then, if Q is the quadratic form on sl 2 (C) induced by the Killing form, the restriction of the bilinear form π * 2 Q to e is some constant multiple of the restriction of π * 1 Q to e. Reversing the assignment of the indices 1, 2 to the summands sl 2 (C) replaces the constant with its reciprocal, so up to this inversion the constant is an invariant of the algebraic model.

D.6 4
∞ Take a 2-sphere and the Euclidean plane. 17 D.6 2 ∞ Take a hyperbolic plane and the Euclidean plane. Up to the joint scaling of pairs of surfaces and local equivalence, this exhausts all of the pairs of constant curvature Riemannian surfaces of unequal curvature.
We can realize some of the remaining real forms with dim h = 6 by extending our attention to rolling distributions generated by pairs of Lorentzian surfaces: We proceed as before, but instead take (Σ 1 , g 1 ), (Σ 2 , g 2 ) to be Lorentzian surfaces, in which case C → Σ 1 × Σ 2 is a principal O(1, 1)-bundle.
With these objects in hand, we can realize the following distributions as follows (and in the first case, with the same restrictions on λ as before): D.6 2+ λ If λ > 0, take two copies of de Sitter space dS 2 (or, just as well, two copies of anti-de Sitter space AdS 2 ) with ratio κ 1 /κ 2 = λ of curvatures. If λ < 0, take a copy of de Sitter space and a copy of anti de Sitter space normalized to have curvature ratio λ. For λ = −1 there are two geometrically distinct possibilities: One for which the copy of de Sitter space has scalar curvature larger than the negative of the scalar curvature of the copy of anti-de Sitter space, and one for which the reverse is true. D.6 1 ∞ Take de Sitter (or anti-de Sitter) space and the Lorentzian plane. One can, of course, use the above identifications as rolling distributions to write down local coordinate realizations for the corresponding models, and this can be done efficiently using the procedure in [2, § 2].
Remark 8. Any two constant curvature surfaces (of the same signature) whose curvatures have ratio 9 : 1 determine a locally flat rolling distribution. Up to replacing both metrics with their negatives, the possibilities are: two spheres, two copies of the hyperbolic plane, and two copies of de Sitter space.
Remark 9. This poses a natural question: Which of the other distributions in the classification (namely, N.7 R r,s , the real forms of N.6 and D.6 * , and D.6 3± −1 ) are realizable as rolling distributions?
Appendix A.
We split cases according to the complex invariant J := 4s/r 2 and, in some cases, the sign of r or s; in each case, one can choose a, b ∈ R satisfying the given equations for r, s. Then, the specified action of F on n −1 5 = s1, s2, t1, t2 together with [F, u] = 0 defines F and thus hr,s := n5 ⋌ E, F . For each entry: kr,s := X, E , where X is the vector in the column labeled kr,s, and dr,s := Y, F ⊕ kr,s, where Y is the vector in the column labeled dr,s. Also, dr,s + [dr,s, dr,s] = Z ⊕ dr,s, where Z is the vector in the column labeled dr,s + [dr,s, dr,s].