Cartan's incomplete classification and an explicit ambient metric of holonomy $\mathrm{G}_2^*$

In his 1910"Five Variables"paper, Cartan solved the equivalence problem for the geometry of $(2, 3, 5)$ distributions and in doing so demonstrated an intimate link between this geometry and the exceptional simple Lie groups of type $\textrm{G}_2$. He claimed to produce a local classification of all such (complex) distributions which have infinitesimal symmetry algebra of dimension at least $6$ (and which satisfy a natural uniformity condition), but in 2013 Doubrov and Govorov showed that this classification misses a particular distribution $\bf E$. We produce a closed form for the Fefferman-Graham ambient metric $\widetilde{g}_{\bf E}$ of the conformal class induced by (a real form of) $\bf E$, expanding the small catalogue of known explicit, closed-form ambient metrics. We show that the holonomy group of $\widetilde{g}_{\bf E}$ is the exceptional group $\textrm{G}_2^*$ and use that metric to give explicitly a projective structure with normal projective holonomy equal to that group. We also present some simple but apparently novel observations about ambient metrics of general left-invariant conformal structures that were used in the determination of the explicit formula for $\widetilde{g}_{\bf E}$.


Introduction
Cartan's most involved application of his powerful equivalence method was the resolution of the equivalence problem for the equivalence problem for the geometry of 2-plane distributions D on 5-manifolds M , which he reported in his landmark 1910 "Five Variables" paper [9]. Such distributions that are maximally nonintegrable in the sense that  [4,6,7] and hence to natural problems in control theory [2], it is intimately linked (as Cartan showed in the aformenetioned article) to the exceptional simple complex Lie group of type G C 2 and then via the natural inclusion G * 2 ֒→ SO (3,4) to conformal geometry [22], and it is the appropriate structure for naturally projectively compactifying strictly nearly (para-)Kähler structures in dimension 6 [17]. In turn, via the Fefferman-Graham ambient construction [13] and a general extension theorem [18] these structures can be used to construct metrics of holonomy contained in and sometimes equal to G * 2 . Cartan claimed to classify locally all (2, 3, 5)-distributions (M, D), implicitly in the complex setting, whose infinitesimal symmetry algebra aut(D) has dimension at least 6 and which satisfies a natural uniformity condition called constant type (see Subsection 2.1). This is the algebra of all vector fields X whose flow preserves D, that is, for which L X Y ∈ Γ(D) for all Y ∈ Γ(D). Some 103 years later, Doubrov and Govorov upended this classification [12] and showed that it misses a single distribution [11], which we denote D ♮ , and which turns out to be a left-invariant distribution on a particular Lie group. (In fact, Doubrov and Govorov reported their missing distribution to the author during his preparation of an earlier article [26] that relied on Cartan's classification; the new distribution was discussed briefly in Example 38 and Remark 39 in that article, but detailed discussion was deferred to the present article. ) Nurowski showed that any (2, 3, 5)-distribution (M, D) canonically induces a conformal structure c D of signature (2, 3) on M [22], and he later showed with Leistner that one can exploit certain distributions of this type to produce explicit ambient metrics g with metric holonomy Hol( g) equal to the split real form G 2 . The ambient metric construction assigns to a conformal structure (M, c) of signature (p, q) an essentially unique Riemannian metric ( M , g) of signature (p + 1, q + 1); see Section 4. Though the ambient metric construction boasts a satisfying existence theorem, finding an ambient metric for a particular conformal structure amounts to solving a formidable nonlinear system of partial differential equations, and consequently explicit closed forms for ambient metrics are known only for a few special classes of conformal structures; we catalogue these in Subsection 4.1. We show that in the special case that c is a left-invariant metric on a Lie group, in a left-invariant frame the ambient metric system reduces to a (still nonlinear) system of ordinary differential equations with constant coëfficients. This is true in particular for the left-invariant conformal structure c ♮ := c D ♮ induced via Nurowski's construction by the exceptional distribution D ♮ (here we abuse notation by using the same notation D ♮ for a particular real distribution, as well as Doubrov and Govorov's missing complex distribution, which is in a precise sense a complexification of that real distribution).
We manage to produce for the conformal structure c ♮ an explicit ambient metric g ♮ whose behavior is qualitatively different from the known explicit examples. Furthermore, via direction computation we show that its holonomy group is equal to G * 2 , furnishing a new example of that uncommonly evidenced exceptional metric holonomy group.
In Section 2, we review some aspects of the geometry of (2, 3, 5)-distributions and present some examples that appear later. We also describe the fundamental curvature invariant for this geometry, introduce a local (quasi)normal form for these geometries, and summarize Nurowski's construction. Section 3 presents Cartan's (purported) classification of highly symmetric complex (2, 3, 5)-distributions and then, in some detail, Doubrov and Govorov's new distribution D ♮ . In Section 4, we review the Fefferman-Graham ambient metric construction, and then we present an inventory of the conformal structures with known explicit ambient metrics. We also give some simple but novel observations regarding the ambient metrics of leftinvariant conformal structures on Lie groups; the author is expanding these ideas in a work in progress [27]. Finally, in Section 5, we ply those observations to produce an explicit ambient metric g ♮ for c ♮ and show that its metric holonomy group is equal to G * 2 . Some computations reported here were performed with the standard Maple Package DifferentialGeometry.
It is a pleasure to thank Mike Eastwood, who encouraged the author to write this example as a standalone paper. It is also a pleasure to thank Boris Doubrov for several relevant (and illuminating) discussions. For reasons including those mentioned in the introduction, the class of distributions that the current article concerns is especially interesting:  [8,24], such 2-plane fields are sometimes instead termed generic.
The geometry of these structures was first studied systematically by Cartan, in his well-known "Five Variables" paper [9]. There, he solved the equivalence problem for this geometry by canonically assigning (locally) to each such distribution (M, D) a principal P -bundle E → M and a g * 2 -valued pseudoconnection ω on E, where P is a particular parabolic subgroup of G * 2 . 1 We will be concerned particularly with (2, 3, 5)-distributions that are highly symmetric; more precisely, we are interested in the dimension of the following algebra: . The Lie algebra aut(D) of all such fields is termed the infinitesimal symmetry algebra of (M, D).
We first describe what one may justifiably call (see Subsection 2.1) a flat model of (2, 3, 5)-distributions.
Let O * be the (8-dimensional) algebra of split octonions, which is endowed with a conjugation map·, and let R and I respectively denote its +1-and (7-dimensional) −1-eigenspaces. Then, the skew-symmetric map × : I × I → I defined by × : (u, v) → ℑ(uv), where juxtaposition denotes split octonion multiplication and ℑ denotes projection onto I with respect to the above eigenspace decomposition, is called the 7-dimensional (split) cross product. We may identify G * 2 as the (linear) automorphism group of (I, ×), that is, the stabilizer of × under the action of GL(I) induced on 2 I * ⊗ I by the standard action; in fact this is the standard (irreducible 7-dimensional) representation of G * 2 . Now, the bilinear form · : I × I → R defined by x · y := − 1 6 tr(z → x × (y × z)) is symmetric and has signature (3,4). Since it is constructed algebraically from ×, it is invariant under the action of G * 2 , and this defines a natural inclusion G * 2 ֒→ SO (3,4). In particular, one can consider the (punctured) null cone C of · ; since G * 2 preserves the bilinear form, it acts on C. Let Π denote the projection of C onto its ray projectivization P + (C), which is homeomorphic to S 2 × S 3 (and which is a double cover of the familiar null quadric variety P(C) ⊂ P(I) defined by the indefinite form · ). For any point u ∈ C, the subspace ker u := {v ∈ I : u × v = 0} ⊂ I is 3-dimensional, by skew-symmetry contains u , and by linearity is invariant under (nonzero) scalings of u; also, for each u we may identify ker u with a subspace of T u I via the canonical isomorphism I ↔ T u I, so that u ⊂ ker T u Π. Thus the 3-plane distribution u → ker u on C descends via T Π to a 2-plane field ∆ on P + (C), and computing gives that it is a (2, 3, 5)-distribution. Since it is defined algebraically in terms of the cross product ×, it is invariant under the G * 2 -action induced on P + (C) by ray projectivization, and in fact aut(∆) ∼ = g * 2 . (See [24] for much more.) Example 2.2. Given the physical system of two (real or complex) oriented surfaces, (Σ 1 , g 1 ) and (Σ 2 , g 2 ), rolling along one another, consider the formal configuration 1 Since G * 2 is semisimple and P is parabolic, this geometry is an example of an important class of structures called parabolic geometries, for which one can canonically encode any structure on a manifold M a bundle E → M and a canonically determined (so-called) Cartan connection; see the standard reference [8] for a general treatment of parabolic geometries and Subsubsection 4.3.2 there for discussion of the geometry of (2, 3, 5)-distributions in this setting. Anyway, as Robert Bryant has observed the pseudoconnection Cartan produces is not in fact a Cartan connection.
space M of all arrangements of the two surfaces tangent at some point, which can be naturally realized as the total space of an SO(2)-bundle P : M → Σ 1 × Σ 2 . For a particular configuration u ∈ M , the base point P (u) = (u 1 , u 2 ) encodes the points u a ∈ Σ a , a = 1, 2, of tangency on each surface, and the fiber over P (u) consists of the relative orientations of the surfaces at the point of tangency, encoded as oriented isometries A : T u1 Σ 1 → T u2 Σ 2 . We can restrict to motions of the system, that is, smooth paths γ(t) = (u 1 (t), u 2 (t), A(t)) in M , without slipping or twisting, which means that These conditions together define on each tangent space T u M 3 independent linear constraints, and hence defines a 2-plane D u ⊂ T u M of permitted infinitesimal motions. These planes together comprise a smooth 2-plane distribution D on M called the rolling distribution for the given pair of surfaces. For a generic pair of surfaces, around a generic point in the configuration space M , there is a neighborhood U to which the restriction of the rolling distribution is a (2, 3, 5)-distribution. For more, see [4,7].
2.1. The fundamental curvature invariant A. The fundamental curvature invariant 2 for the geometry of (2, 3, 5)-distributions (M, D) turns out to be a field of binary quartic forms defined on the given distribution D, that is, a section of A ∈ Γ( 4 D * ) [9, §6]; its value at a point depends only on the 6-jet of D at that point. Akin to the Riemann curvature tensor in Riemannian geometry, A is a complete obstruction in the sense that it is zero everywhere iff D is locally equivalent at each point to the (hence named) flat distribution ∆ in Example 2.1. If D is complex, then at any point u ∈ M either A u = 0 or A u has exactly four roots counting multiplicity. In the latter case we call the partition of 4 given by the multiplicities the root type of D at u, and if A u = 0 we say that D has root type [∞] at u. If the root type of D u is Λ at every u ∈ M , then we simply say that D itself has constant root type Λ.
If D is real, we can consider the complexified form A ⊗ C ∈ Γ( 4 (D ⊗ C) * ), and then define root type and constant root type in terms of that form.

2.2.
A quasinormal form. This geometry admits a local quasi-normal form: Any ordinary differential equation can be regarded as a differential system on the partial jet space where the variables p and q are placeholders respectively for y ′ and y ′′ , comprising the canonical forms dy − p dx, dp − q dx, and the particular form In particular, these forms are linearly independent at each point, and so their common kernel D F is a 2-plane distribution on F 5 xypqz , namely, Concretely, a pair (y(x), z(x)) is a solution of (1) iff its prolongation is an integral curve of D F . Computing directly shows that D F is a (2, 3, 5)-distribution iff ∂ 2 q F vanishes nowhere. Goursat showed that, in fact, locally every (2, 3, 5) distribution arises this way: A distribution D F given by a function F is said to be in Monge normal form.
2.3. The canonical conformal structure. Nurowski showed that any real (2, 3, 5)distribution determines a canonical conformal structure on the underlying manifold: There is a natural functor that assigns to a (2, 3, 5)distribution (M, D) a conformal structure c D of signature (2, 3) on M . The distribution is totally null with respect to c D , and the value of the conformal structure at a point depends only on the 4-jet of D at that point.
Proof. (Sketch.) Proceeding nearly as Cartan did, we can associate to (M, D) a unique Cartan connection ω : E → g * 2 on a principal P -bundle E → M , where P is a particular parabolic subgroup of G * 2 , namely the stabilizer of a fixed element u of the ray projectivization of the null cone P + (C) described in Example 2.1. Now, the group SO(3, 4) canonically associated to G * 2 likewise acts on P + (C), and letP denote the stabilizer of u under this action; by construction, P =P ∩ G * 2 , so we can form the extended bundleĒ = E × PP → M and extend the Cartan connection ω equivariantly to a Cartan connectionω :Ē → so (3,4). Butω is exactly [19, Proposition 4] Cartan's normal connection associated to some (unique) conformal structure on M [10]. (In fact, we can only assign a connection ω to an oriented (2, 3, 5)-distribution; however, the construction is local and so in the nonorientable case we may apply it to a cover by oriented neighborhoods and patch the resulting conformal structures into a global conformal structure.) If we replace groups appropriately, the proof applies just as well to show that a complex (2, 3, 5)-distribution induces a complex conformal structure on the underlying manifold.
Nurowski also gave an explicit (and, with a length of about 60 terms, remarkable and daunting) formula for a representative of the conformal structure c DF of a distribution D F in Monge normal form [22, §5.3]; its coëfficients with respect to the coördinate basis are polynomials of degree at most 6 in the 4-jet of F .
One can compute the fundamental curvature invariant A of Subsection 2.1 explicitly in terms of the natural conformal structure c D . The filtration D ⊂ [D, D] ⊂ T M defined by the underlying distribution D determines a lattice of subspaces of the space of algebraic Weyl tensors at each point of M . The Weyl curvature of the conformal distribution D projects naturally onto a bundle associated to a particular subquotient associated to this lattice isomorphic to 4 D * , and we can identify the image of the Weyl curvature under this projection with A; see [18] for a more detailed description of this computation.
3. The missing distribution D ♮ 3.1. Cartan's ostensible classification. Cartan claimed to classify, up to local equivalence, and implicitly in the complex setting, all (2, 3, 5)-distributions whose infinitesimal symmetry algebra has dimension at least 6 and which have constant root type. The structures he found were the following: [9, § §8-11] • The rolling distributions D ρ for a pair of spheres whose radii have ratio ρ > 1. In particular, these distributions admit infinitesimal symmetries corresponding to arbitrary independent infinitesimal rotations of the spheres, and so aut(D ρ ) ≥ so(3, C) × so(3, C) and hence dim aut(D r ) ≥ 6. In fact, equality holds (and the root type of D ρ is [2,2]) for all ratios except the critical value of ρ = 3. In that case, the distribution D 3 is locally equivalent at each point to the flat model distribution ∆ and hence has root type [∞] (in fact, P(C + ) is a double-cover of the space underlying D 3 , and ∆ is the pullback of D 3 by the covering map), so that aut(D 3 ) ∼ = g * 2 and dim aut(D 3 ) = 14. See [1,5,6,7,28] for much more.
• The rolling distribution D ∞ for a sphere and a plane. The symmetry algebra aut(D ∞ ) is isomorphic to the affine Lie algebra so(3, C) ⋌ C 3 , and the root type is [2,2]. • A distribution D ′ with symmetry algebra aut(D ′ ) ∼ = so(3, C) × (so(2, C) ⋌ C 2 ) and root type [2,2]. 3.2. Doubrov-Govorov's example. Some 103 years after Cartan's work, Doubrov and Govorov classified the homogeneous (complex) (2, 3, 5)-distributions [11] and found among them a single distribution D ♮ that has 6-dimensional infinitesimal symmetry algebra but which is missing from Cartan's classification: Define a Lie algebra structure g ♮ on a 5-dimensional vector space over F (F = R or F = C) with basis (E a ) by the bracket relations Then, let G ♮ be a connected Lie group with Lie algebra g ♮ and let D ♮ be the left-invariant 2-plane field determined by D ♮ 0 . Via a usual abuse of notation, let (E a ) also denote the left-invariant frame whose restriction to id ∈ G ♮ is the basis (E a ) ⊂ T id G ♮ ∼ = g ♮ ; then, computing using the above bracket relations gives and so D ♮ is a (2, 3, 5)-distribution. One can show (for example, by computing the conformal structure given below and then projecting the resulting Weyl curvature onto the appropriate subquotient as described in Subsection 2.3) that D ♮ has constant root type [4]. Doubrov and Govorov integrated these structure equations and produced a Monge normal form D F ♮ for the distribution, determined by the function F ♮ (x, y, p, q, z) := y + q 1/3 , on a suitable domain (say, on {q > 0} in the real case, and on a compliment of a branch cut for q → q 1/3 in the complex case). For aesthetic reasons, we change to coördinates (x, y, p, r, z), where q = r 3 .
One can verify directly that this is a Monge normal form for D ♮ by showing that the frame L a defined by satisfies the bracket relations of (E a ) and the identifications E a ↔ L a identify D ♮ with D F ♮ (we make these identifications henceforth).
Since it is left-invariant, the infinitesimal symmetry algebra aut(D ♮ ) contains as a Lie subalgebra the space of right-invariant vector fields on G ♮ ; one basis of this subalgebra is (R a ), where Direct computation shows that the vector field R 6 := −y∂ x + p 2 ∂ p + pr∂ r − 1 2 y 2 ∂ z , which is not right-invariant, is also in aut(D ♮ ), which hence has dimension at least 6. Tedious verification shows that this field and the subalgebra of right-invariant fields together span the full algebra of infinitesimal automorphisms, and then that aut(D ♮ ) ∼ = sl(2, F) ⋌ n 3 , where n 3 is the 3-dimensional Heisenberg algebra over F. Proof. The algebra aut(D ♮ ) is isomorphic to sl(2, C) ⋌ n 3 , but this is not the infinitesimal symmetry algebra of any distribution in Cartan's list.
Alternatively, D ♮ has constant root type [4], and its symmetry algebra contains sl(2, C) as a subalgebra and hence is insoluble. The infinitesimal symmetry algebras of all of the distributions on Cartan's list that have constant root type [4], however, are solvable.
In the left-invariant coframe (e a ) on G * dual to (E a ), the conformal structure c ♮ := c D ♮ induced by the real distribution D ♮ contains the left-invariant representative metric (3) g ♮ := −2(e 1 ) 2 − 2e 1 e 3 + (e 2 ) 2 + 4e 2 e 5 + 4e 3 e 4 .   [4] and that its infinitesimal symmetry algebra is spanned by the right-hand side of (4) and One can show that this distribution is locally equivalent to D ♮ (when both distributions are regarded as real, and when both are regarded as complex).

Ambient metrics
The Fefferman-Graham ambient construction associates to a conformal structure (M, c) of signature (p, q), dim M = p + q ≥ 2, a unique pseudo-Riemannian metric ( M , g) of signature (p+1, q+1). When dim M is odd, this construction is essentially unique, and hence invariants of g (that are independent of the choices made) are also invariants of the underlying conformal structure; indeed, this was the original motivation for the construction [13].
We describe the ambient metric construction for odd dimensions following [13]; the discussion here is similar to that in [18], which also includes details of the more subtle case when n ≥ 4 is even.
The metric bundle associated to c is the ray bundle G → M defined by The bundle G enjoys a natural topology and smooth structure so that its smooth sections are precisely the representative metrics of c. Now, the natural dilation action R + × G → G defined by s · g x = δ s (g x ) := s 2 g x realizes G as a principal R + -bundle. The metric bundle also admits a tautological symmetric 2-tensor g 0 ∈ Γ( 2 T * G) defined by it annihilates π-vertical vectors and hence is degenerate. By construction, δ * s g 0 = s 2 g 0 .
Fixing a representative g ∈ c yields a trivialization G ↔ R + × M that identifies t 2 g x ↔ (t, x). With respect to this trivialization, the tautological 2-tensor is given by g 0 = t 2 π * g and the dilations by δ s : (t, x) → (st, x). Now, consider the space G × R and denote the standard coördinate on R by ρ. Then, the map ι : G ֒→ G × R defined by z → (z, 0) embeds G as a hypersurface in G × R and we identify G with its image. The dilations δ s extend trivially to G × R, that is, by s · (g x , ρ) = δ s (g x , ρ) := (s 2 g x , ρ). Likewise, a choice of representative g ∈ c determines a trivialization G×R ↔ R + ×M that identifies (t 2 g x , ρ) ↔ (t, x, ρ), which in turn defines an embedding M ֒→ G × R by (5) x → (1, x, 0) and yields an identification (1) it extends g 0 in the sense that ι * g = g 0 , and (2) it has the same homogeneity with respect to the dilations δ s as g 0 , that is, if δ * s g = s 2 g. A pre-ambient metric is straight if for all u ∈ M the parameterized curve R + → M defined by u → u · p is a geodesic. Any nonempty conformal structure admits many pre-ambient metrics; Cartan's normalization condition for a conformal connection [10] suggests that Ricci-flatness is a natural distinguishing criterion. Here, we say that a tensor field on M is O(ρ ∞ ) if it vanishes to infinite order in ρ at each point in the zero set G of ρ.
We formulate Fefferman-Graham's existence and uniqueness results for ambient metrics of odd-dimensional conformal structures as follows: The proof of the theorem uses the fact that for any representative g ∈ c we may write any ambient metric g (possibly after restricting to an open, dilation-invariant set containing G) in the normal form g = 2ρ dt 2 + 2t dt dρ + t 2 g(x, ρ) where g(x, 0) = g and where we may regard g(x, ρ) as a family of metrics on M depending on the parameter ρ; here and henceforth we suppress notation for pullback by the inclusion M ֒→ M determined by g.
For g in this form, decomposing the equation Ric( g) = 0 with respect to the splitting where g = g(x, ρ), ′ denotes differentiation with respect to ρ, and ∇ a and R ab respectively denote the Levi-Civita connection and Ricci curvature of g(x, ρ) with ρ fixed. The components R tt , R ta , and R tρ are identically zero. This system and the initial condition g(x, 0) = g together comprise a normal-form initial value problem. Now, differentiating these expressions, setting them equal to zero, and then evaluating at ρ = 0 successively determines all of the derivatives g (m) (x, ρ): (Still for n odd,) the equation for R ab alone determines ∂ (m) ρ g ab for m < n and the tracefree part of ∂ (n) ρ g ab . The equation for R ρρ determines the trace part g ab ∂ (n) ρ g ab , and then the equation for R ab determines all higher derivatives. If g is real-analytic, then the power series of g along G (at ρ = 0) converges to a real-analytic ambient metric g on some open subset of M containing G, and in particular Ric( g) = 0.

Explicit ambient metrics: A brief catalogue.
Despite that ambient metrics always exist, explicit, closed-form ambient metrics have been produced for only a few isolated classes of conformal structures, owing in part to the severe nonlinearity of the system (6) of partial differential equations. We list here the (published) examples of which the author is aware (as well as an example involving a compact power series that may not terminate). In principle it is easy to verify that the metrics indicated here are indeed ambient metrics: One needs only to check that they are Ricci-flat, at least to infinite order in ρ, and that their pullback via the inclusion ι is g 0 , or in the case that the ambient metric is in normal form with respect to some representative metric g, that the pullback is t 2 g. In all cases here, the underlying manifold is taken to have dimension n ≥ 3.  ) is Einstein, that is, if c admits an Einstein representative g, say, with R ab = 2λ(n − 1)g ab , λ ∈ R, then the (normal form) metric g = 2ρ dt 2 + 2t dt dρ + t 2 (1 + λρ) 2 g is an ambient metric for c.
In practice one can often produce explicit ambient metrics for so-called almost Einstein conformal structures [15], for example, for the conformal structures induced by Cartan's distributions D I described in Subsection 3.1 [26,Proposition 27].
where we suppress the notation for the pullbacks by the projections M → M i [16,Theorem 2.1]. They also describe how to extend this construction to products of more than two Einstein factors. Example 4.3. A pp-wave 3 is a Lorentzian metric (M, g) of dimension n that admits a nonzero parallel null vector field. For any such vector field, a pp-wave admits around any point coördinates (x 1 , . . . , x n−2 , u, r) such that the vector field has representation ∂ r and the metric has representation for some function h. It is Einstein (in fact, Ricci-flat, since any pp-wave is scalarflat) iff ∆h = 0, where Leistner and Nurwoski showed that if n is odd or n is even and ∆ n/2 h = 0, then is an ambient metric for the conformal class [g] containing the pp-wave, where P k := k j=1 (2j − n). If n is even, this metric is polynomial in ρ of degree n 2 − 1; if n is odd, the above series converges on some interval in ρ. [21] Example 4.4. Perhaps the most remarkable class of examples is also due to Nurowski [23]. Consider the 8-parameter family of (2, 3, 5)-distributions given in Monge normal form D F [a;b] by the function where a 0 , . . . , a 6 , b are real constants. Generically, the formula for the conformal structure c D F [a;b] has about 30 terms, so we do not produce it here. This class is interesting for at least two reasons: First (and perhaps surprisingly), the ambient metrics g F [a;b] are (at most) quadratic polynomials in ρ. Second, as Leistner and Nurowski showed, for generic parameter values (in fact, precisely when at least one of a 3 , a 4 , a 5 , a 6 is nonzero, which is the case exactly when D F [a;b] has root type [3,1] at generic points), the metric holonomy of g F [a;b] is G * 2 , so this gives a construction for some metrics with that uncommonly evidenced metric holonomy group.

Ambient metrics of left-invariant conformal structures.
The generally difficult problem of computing explicit ambient metrics of conformal structures in principle simplifies significant in the special case of left-invariant conformal structures (G, c) on Lie groups. We indicate here some of these simplifications but take up a more detailed treatment later in a work in progress.
First, a left-invariant conformal structure c trivially enjoys a distinguished line of representative metrics, namely those that are themselves left-invariant, and the problem of finding an explicit expression for the ambient metric simplifies in a critical way when writing the ambient metric in normal form with respect to such a metric. Given any left-invariant frame (E a ) of T G, the Lie algebra g of G is described by the structure constants C c ab characterized by Lie bracket relations [E a , E b ] = C c ab E c , and with respect to that frame the left-invariant metric g ∈ c decomposes as g ab e a ⊗ e b , where (e a ) is the (left-invariant) coframe dual to (E a ), for constants g ab . So, any local invariant of the metric is left-invariant, and in particular the quantities ∇ Ei E j are left-invariant and the components R ab of Riccicurvatures are constants (in fact, they are quadratic combinations of the structure constants). Then, an inductive argument (using the method for computing derivatives g (m) (x, 0) described after (6) in Section 4) gives that the derivatives g (m) (x, 0) are all local invariants of the metric, which hence are themselves left-invariant. Thus, g(x, ρ) is itself left-invariant for each ρ, and hence g is invariant under the trivial extension of the left action of G to the ambient space G ⊆ R + × G × R. In particular, the components g ab of the metric g(x, ρ), regarded as a family of metrics on M parameterized by ρ, are functions g ab (ρ) of ρ alone. These observations together imply substantial simplifications of the initial value problem in this setting, which we summarize as follows: Proposition 4.1. Let (G, c) be a left-invariant conformal structure on a Lie group and g ∈ c a left-invariant representative metric. Then, interpreted as a system of equations on the coëfficients g ab of g(x, ρ) with respect to a left-invariant frame, the system (6) of partial differential equations characterizing the components of an ambient metric in normal form reduces to a system of ordinary differential equations for g ab in ρ with constant coëfficients.

A new explicit ambient metric
With some effort, one can produce an explicit ambient metric for the conformal structure (G ♮ , c ♮ ): Proposition 5.1. In the notation of Subsection 3.2, the metric is an ambient metric for the (real) left-invariant conformal structure (G ♮ , c ♮ ) induced by the (real) distribution D ♮ .
Proof. Evaluating the quantity in parentheses at ρ = 0 shows that g ♮ is in normal form with respect to g ♮ (3), and computing directly shows that Ric( g ♮ ) = 0. Since solving the system (6) is typically intractable, we indicate briefly our naïve method for producing the explicit solution (7). First, choosing a left-invariant frame (E a ) with respect to which both the left-invariant representative metric g ab and the bracket relations are particularly simple yields a simpler initial value problem: A simpler collection of structure constants (C c ab ) roughly yields simpler Ricci curvature components R ab , simplifying the first equation in the system, and a simpler metric g ab gives a simpler initial condition. With relatively simple coëfficients g ab and R ab in hand, one can proceed with the algorithm described after (6) to determine successive derivatives g (m) ab (0). In our case, we observe after computing the first several derivatives we observe some patterns: First, only certain components g ab (ρ) have nonzero derivatives among them (in our case, up to the symmetry g ab = g ba , only g 11 , g 13 , g 25 , g 33 , g 34 ).
The truncated Taylor series at ρ = 0 agree with the those of functions of a familiar form, and using these functions as coëfficients yields the candidate normal-form ambient metric g ♮ (7). Computing gives that Ric( g ♮ ) = 0, so this is an ambient metric for c ♮ as desired.
5.1. G * 2 holonomy. Here we show that the metric g ♮ produced in the previous subsection has holonomy equal to the split real form G * 2 of the exceptional complex simple Lie group G C 2 . We use the following theorem, which is an application of a much broader "tractor extension theorem" [18, Theorem 1.4]: Theorem 5.1. [18, Theorem 1.1] Let D be an oriented, real-analytic (2, 3, 5)distribution. Then, the metric holonomy Hol( g D ) of any real-analytic ambient metric g D for the conformal structure c D is contained in G * 2 . Proposition 5.2. The metric holonomy Hol( g ♮ ) of the ambient metric ( G ♮ , g ♮ ) is equal to G * 2 . Proof. By Theorem 5.1, Hol( g ♮ ) ≤ G * 2 , so to prove the claim, it suffices to show that the dimension of the holonomy group (or equivalently its Lie algebra) is equal to dim G * 2 = 14. (In fact, since the maximal subgroups of G 2 are the 9-dimensional parabolic subgroups, it suffices to show that the dimension of the holonomy group is greater than 9.) By the Ambrose-Singer Theorem [3], the Lie algebra of Hol( g ♮ ) contains (in fact, since g ♮ is real-analytic, is equal to) the infinitesimal holonomy at any point u, which is the algebra hol u ( g ♮ ) ⊆ so(T u G ♮ ) generated by the value of its curvature R and the derivatives thereoef at a point u, or more precisely, the endomorphisms where ∇ is the Levi-Civita connection of g ♮ . Computing gives that the image of R u in End(T u G ♮ ) (where the basepoint u ∈ G ♮ , which we suppress below, is any point with t = 1, ρ = 0), is spanned by the linearly independent endomorphisms R(E 1 , E 2 ), R(E 1 , E 4 ), R(E 1 , ∂ ρ ), R(E 2 , E 3 ), R(E 2 , E 4 ), R(E 2 , ∂ ρ ), and that those elements together with form a basis of the space of endomorphisms generated by at most one derivative of R. But this basis has dim G * 2 = 14 elements, so Hol( g ♮ ) = G * 2 .